Solving Math Problems Step By Step Guide

In this article, we will delve into solving several mathematical expressions, providing step-by-step solutions and explanations. We will cover topics such as simplifying expressions with exponents, dealing with negative exponents, working with radicals, representing numbers on a number line, and more. This guide aims to provide a clear and concise understanding of the methods involved in solving these problems.

1) Simplifying Expressions with Fractional Exponents: a^(1/3) (a^3 - a^(5/2))

When dealing with expressions involving fractional exponents, it's crucial to understand the rules of exponents. Our focus in this section is simplifying the expression a^(1/3) (a^3 - a^(5/2)). This expression combines fractional exponents and the distributive property, requiring a careful application of exponent rules to arrive at the simplified form. Let's break down the process step by step.

First, we need to distribute the a^(1/3) term across the terms inside the parenthesis. This involves multiplying a^(1/3) by both a^3 and a^(5/2). Remember, when multiplying terms with the same base, you add their exponents. This principle is key to simplifying the expression correctly. The initial distribution sets the stage for applying the exponent rules, making the subsequent steps more manageable.

Next, we apply the rule of adding exponents when multiplying terms with the same base. For the first term, we add the exponents 1/3 and 3. To do this, we need a common denominator, so we rewrite 3 as 9/3. Adding 1/3 and 9/3 gives us 10/3. Thus, the first term simplifies to a^(10/3). For the second term, we add the exponents 1/3 and 5/2. The common denominator for 3 and 2 is 6, so we rewrite 1/3 as 2/6 and 5/2 as 15/6. Adding 2/6 and 15/6 gives us 17/6. So, the second term simplifies to a^(17/6). This step-by-step approach ensures accuracy in handling fractional exponents.

Therefore, the simplified expression is a^(10/3) - a^(17/6). This result combines the simplified forms of both terms after applying the distributive property and the rules of exponents. The expression is now in its most basic form, demonstrating the application of exponent rules in simplifying algebraic expressions.

2) Simplifying Expressions with the Same Base: (x^2 / x²⁾(2/3)

In this section, our aim is to simplify the expression (x^2 / x²⁾(2/3). This expression involves dividing terms with the same base and then raising the result to a fractional exponent. Understanding the order of operations and the rules of exponents is essential for simplifying this expression correctly. The step-by-step approach outlined below will guide you through the process.

First, we simplify the expression inside the parentheses. We have x^2 divided by x^2. According to the rules of exponents, when you divide terms with the same base, you subtract the exponents. In this case, we subtract 2 from 2, which equals 0. Therefore, x^2 / x^2 simplifies to x^0. This initial simplification is crucial for making the expression more manageable.

Next, we know that any non-zero number raised to the power of 0 is 1. So, x^0 equals 1. Now our expression becomes (1)^(2/3). This simplification makes the next step straightforward.

Finally, we raise 1 to the power of 2/3. Any root or power of 1 is always 1. Therefore, (1)^(2/3) simplifies to 1. This final step completes the simplification process, giving us the final answer.

Thus, the simplified form of the expression (x^2 / x²⁾(2/3) is 1. This result highlights the importance of following the rules of exponents and the order of operations when simplifying mathematical expressions. The clear, step-by-step simplification ensures an accurate and understandable solution.

3) Working with Negative Exponents: (-3)^3 ⋅ (-3)^(-4) ⋅ (-3)^(-3)

Dealing with negative exponents requires a solid understanding of exponent rules. The problem we're addressing here is (-3)^3 ⋅ (-3)^(-4) ⋅ (-3)^(-3). This expression involves multiplying terms with the same base but with varying positive and negative exponents. The key to simplifying this is to apply the rule that states when multiplying terms with the same base, you add the exponents. Let's break this down step by step.

First, we need to add the exponents. We have 3, -4, and -3. Adding these together, we get 3 + (-4) + (-3) = 3 - 4 - 3 = -4. This means our expression will simplify to (-3)^(-4). Combining the exponents in this way is a fundamental step in simplifying expressions with multiple exponents.

Next, we need to deal with the negative exponent. A term raised to a negative exponent is the same as the reciprocal of the term raised to the positive exponent. In other words, a^(-n) = 1 / a^n. Applying this rule, (-3)^(-4) becomes 1 / (-3)^4. This step transforms the expression into a more manageable form for calculation.

Finally, we evaluate (-3)^4. This means multiplying -3 by itself four times: (-3) ⋅ (-3) ⋅ (-3) ⋅ (-3). Since a negative number raised to an even power is positive, we get 81. Thus, our expression becomes 1 / 81. This final calculation gives us the simplified form of the original expression.

Therefore, the simplified form of (-3)^3 ⋅ (-3)^(-4) ⋅ (-3)^(-3) is 1/81. This result demonstrates the correct application of exponent rules, particularly those involving negative exponents. The step-by-step approach ensures a clear and accurate simplification process.

4) Simplifying Expressions with Radicals and Exponents: (3√(-8))^9

This problem involves simplifying an expression with both radicals and exponents: (3√(-8))^9. Dealing with radicals, especially with negative numbers under the radical, requires careful attention to detail and a good understanding of complex numbers. The step-by-step approach below will guide you through the simplification process, ensuring a clear and accurate solution.

First, we need to simplify the cube root of -8. The cube root of -8 is -2 because (-2) ⋅ (-2) ⋅ (-2) = -8. So, our expression inside the parentheses becomes 3 ⋅ (-2). This step simplifies the radical part of the expression.

Next, we multiply 3 by -2, which gives us -6. Now our expression inside the parentheses is simplified to -6. So, the entire expression becomes (-6)^9. This simplification makes the exponentiation step easier to handle.

Finally, we need to evaluate (-6)^9. This means multiplying -6 by itself nine times. Since we are raising a negative number to an odd power, the result will be negative. Calculating this, we get -10,077,696. This final calculation provides the simplified value of the original expression.

Therefore, the simplified form of (3√(-8))^9 is -10,077,696. This result showcases the importance of simplifying radicals and applying exponent rules correctly. The step-by-step approach ensures that each part of the expression is handled accurately, leading to the correct final answer.

5) Representing Fractions on a Number Line: -17/3

Representing a fraction on a number line involves understanding the value of the fraction and its position relative to whole numbers. Our task here is to represent -17/3 on a number line. This fraction is an improper fraction, meaning the numerator is greater than the denominator, and it's also negative, which means it will be located to the left of zero on the number line. Let’s break down the process step by step.

First, we need to convert the improper fraction -17/3 into a mixed number. To do this, we divide 17 by 3. The quotient is 5, and the remainder is 2. Therefore, -17/3 can be written as -5 2/3. This conversion helps us visualize the location of the fraction on the number line more easily.

Next, we know that -5 2/3 lies between -5 and -6 on the number line. It's important to understand that negative numbers increase in value as they approach zero, so -5 2/3 is greater than -6 but less than -5. This positioning is crucial for accurate representation.

Finally, to accurately place -5 2/3 on the number line, we divide the space between -5 and -6 into three equal parts, since the fraction part is 2/3. We then count two parts from -5 towards -6. This point represents -5 2/3 or -17/3 on the number line. This final step gives a precise visual representation of the fraction.

Therefore, representing -17/3 on a number line involves converting it to a mixed number, identifying the whole numbers it lies between, and dividing the space accordingly. This process provides a clear visual understanding of the fraction's value and position.

6) Simplifying Expressions with Radicals: (5√(6000)) / (5√6)

Simplifying radical expressions often involves identifying perfect squares within the radical and using properties of radicals to combine or simplify terms. The expression we are tackling here is (5√(6000)) / (5√6). This expression involves dividing two radical terms, and the key to simplifying it is to manipulate the radicals to find common factors. Let’s go through the simplification process step by step.

First, we can simplify the radicals by factoring out perfect squares from under the square root. The number 6000 can be factored as 6 ⋅ 1000, and 1000 can be further factored as 100 ⋅ 10. Since 100 is a perfect square, we can rewrite √(6000) as √(6 ⋅ 100 ⋅ 10), which simplifies to 10√(6 ⋅ 10) or 10√60. So, the expression becomes (5 ⋅ 10√60) / (5√6). This factoring step is crucial for simplifying radicals.

Next, we can simplify √60 further. 60 can be factored as 6 ⋅ 10, so √60 can be written as √(6 ⋅ 10). This does not directly give us a perfect square, but it helps us see the common factor of 6 more clearly. Our expression is now (5 ⋅ 10√(6 ⋅ 10)) / (5√6). Recognizing these factors helps in the next step of simplification.

Then, we can simplify the expression by canceling out common factors. The number 5 appears in both the numerator and the denominator, so we can cancel them out. The expression now becomes (10√(6 ⋅ 10)) / √6. Next, we rewrite √(6 ⋅ 10) as √6 ⋅ √10. Our expression is now (10 ⋅ √6 ⋅ √10) / √6. We can cancel out the √6 term from both the numerator and the denominator. This cancellation simplifies the expression significantly.

Finally, after canceling out the common factors, we are left with 10√10. This is the simplest form of the expression. The step-by-step simplification process has allowed us to reduce the complex expression to a more manageable form.

Therefore, the simplified form of (5√(6000)) / (5√6) is 10√10. This result demonstrates the importance of factoring out perfect squares and canceling common factors when simplifying radical expressions. The methodical approach ensures an accurate and understandable solution.

7) Evaluating Expressions with Fractional Exponents: (25/16)^(1/2)

Evaluating expressions with fractional exponents involves understanding that a fractional exponent represents a root. The problem we are going to solve is (25/16)^(1/2). The exponent 1/2 indicates that we need to find the square root of the fraction inside the parentheses. Let’s break down the process step by step.

First, we recognize that the exponent 1/2 means taking the square root. So, (25/16)^(1/2) is the same as √(25/16). This understanding is crucial for proceeding with the simplification. Converting the fractional exponent to a radical makes the problem easier to visualize.

Next, we apply the property that the square root of a fraction is the square root of the numerator divided by the square root of the denominator. In other words, √(a/b) = √a / √b. Applying this to our expression, we get √25 / √16. This step separates the problem into finding two individual square roots.

Then, we evaluate the square root of 25 and the square root of 16. The square root of 25 is 5 because 5 ⋅ 5 = 25, and the square root of 16 is 4 because 4 ⋅ 4 = 16. So, our expression simplifies to 5/4. This calculation gives us the simplified value of the square roots.

Therefore, the simplified form of (25/16)^(1/2) is 5/4. This result demonstrates how to handle fractional exponents by converting them to radicals and applying the properties of square roots. The step-by-step approach ensures a clear and accurate evaluation.

8) Simplifying Expressions with Cube Roots: 5∛(x³/y³) × 5

Simplifying expressions involving cube roots and variables requires a solid understanding of radical properties. Our goal here is to simplify 5∛(x³/y³) × 5. This expression combines a cube root with variables raised to powers and involves multiplication. Let's break down the simplification process step by step.

First, we focus on simplifying the cube root part of the expression, which is ∛(x³/y³). Recall that the cube root of a fraction is the cube root of the numerator divided by the cube root of the denominator, i.e., ∛(a/b) = ∛a / ∛b. Applying this, we get ∛(x³) / ∛(y³). This separation allows us to simplify the cube roots individually.

Next, we evaluate the cube roots of x³ and y³. The cube root of x³ is x, because x ⋅ x ⋅ x = x³, and the cube root of y³ is y, because y ⋅ y ⋅ y = y³. So, our expression becomes x/y. This simplification step is crucial for reducing the complexity of the expression.

Then, we substitute this simplified form back into the original expression. Our expression is now 5 ⋅ (x/y) ⋅ 5. This step brings us closer to the final simplified form by replacing the cube root term with its simplified equivalent.

Finally, we multiply the constants together. We have 5 ⋅ 5, which equals 25. So, the simplified expression is 25(x/y) or 25x/y. This final calculation completes the simplification process, giving us a clear and concise result.

Therefore, the simplified form of 5∛(x³/y³) × 5 is 25x/y. This result showcases the importance of understanding and applying the properties of cube roots and simplifying expressions step by step. The methodical approach ensures accuracy and clarity in the solution.

Conclusion

In this comprehensive guide, we've tackled various mathematical expressions, providing detailed, step-by-step solutions. We've covered simplifying expressions with fractional and negative exponents, working with radicals, representing fractions on a number line, and more. By understanding the rules and properties involved in each type of problem, you can confidently approach and solve similar mathematical challenges. Remember, the key is to break down complex problems into smaller, manageable steps and apply the appropriate rules and properties at each stage. This methodical approach not only leads to accurate solutions but also enhances your overall understanding of mathematical concepts.