Multiplying And Dividing Exponential Expressions Solving For Unknowns

Understanding Exponential Expressions

Before diving into solving these problems, let's solidify our understanding of exponential expressions, particularly those with negative exponents. A negative exponent indicates the reciprocal of the base raised to the positive value of the exponent. In simpler terms, x^(-n) is equivalent to 1/(x^n). This foundational concept is crucial for manipulating and simplifying the expressions in our problems.

When we encounter expressions like (-7)^(-2) or (-21)^(-3), it's essential to apply this rule diligently. For instance, (-7)^(-2) translates to 1/((-7)^2), which further simplifies to 1/49. Similarly, (-21)^(-3) becomes 1/((-21)^3), resulting in 1/(-9261). These transformations are not just about changing the form of the expression; they are about revealing the true value and nature of the number.

The base of an exponential expression can also be a fraction, as seen in (-6/13)^(-3). The rule for negative exponents still applies, but we also need to remember how to raise a fraction to a power. When a fraction (a/b) is raised to a power n, both the numerator and the denominator are raised to that power: (a/b)^n = a^n / b^n. When the exponent is negative, we also take the reciprocal of the fraction. Thus, (-6/13)^(-3) is equivalent to (13/-6)^3. This inversion is key to simplifying the expression and making it easier to work with in subsequent calculations.

Understanding the interplay between negative exponents and fractional bases is fundamental to mastering these types of mathematical problems. The ability to fluently convert between these forms allows us to approach complex questions with confidence and clarity. This groundwork is not just for solving specific problems; it builds a stronger foundation for more advanced mathematical concepts.

Problem 1: Finding the Multiplier for (-7)^(-2) to Obtain (-21)^(-3)

Our first task involves determining the number by which we must multiply (-7)^(-2) to arrive at (-21)^(-3). To methodically solve this, let's denote the unknown number as 'x'. This allows us to set up an equation that precisely represents the problem's condition. The equation is: x * (-7)^(-2) = (-21)^(-3). This equation encapsulates the core of the question, transforming the word problem into a concrete mathematical statement.

The next step is to simplify the exponential expressions. We've already established that a negative exponent signifies the reciprocal of the base raised to the positive exponent. Applying this, (-7)^(-2) becomes 1/((-7)^2), which equals 1/49. Similarly, (-21)^(-3) is equivalent to 1/((-21)^3), which equals -1/9261. Substituting these simplified forms into our equation, we get: x * (1/49) = -1/9261. This substitution makes the equation more manageable and reveals the numerical relationships more clearly.

To isolate 'x' and find its value, we need to perform an algebraic manipulation. We multiply both sides of the equation by 49, which is the reciprocal of 1/49. This operation cancels out the fraction on the left side, leaving 'x' by itself. On the right side, we perform the multiplication: 49 * (-1/9261). This results in x = -49/9261. To express this fraction in its simplest form, we look for common factors between the numerator and the denominator. Both 49 and 9261 are divisible by 49. Dividing both by 49, we simplify the fraction to x = -1/189. Therefore, the number we need to multiply (-7)^(-2) by to obtain (-21)^(-3) is -1/189.

This step-by-step approach, from setting up the equation to simplifying and solving for 'x', showcases the importance of methodical problem-solving. It highlights how understanding the properties of exponents, combined with algebraic manipulation, can lead us to the solution.

Problem 2: Finding the Divisor for (-6/13)^(-3) to Obtain (-12/39)^(-3)

In this part, we are tasked with finding the number by which we should divide (-6/13)^(-3) to get (-12/39)^(-3). Let's denote this unknown divisor as 'y'. Constructing an equation to represent this scenario, we have: (-6/13)^(-3) / y = (-12/39)^(-3). This equation clearly outlines the mathematical relationship we need to solve.

Before we can solve for 'y', we need to simplify the expressions involving negative exponents. Recall that a negative exponent implies taking the reciprocal of the base raised to the positive exponent. Applying this rule, (-6/13)^(-3) becomes (13/-6)^3. To further simplify, we raise both the numerator and the denominator to the power of 3: (13^3) / (-6^3) = 2197 / -216. Similarly, (-12/39)^(-3) can be rewritten as (39/-12)^3. Calculating the cubes, we get (39^3) / (-12^3) = 59319 / -1728. However, we can simplify the fraction -12/39 before raising it to the power. Both -12 and 39 are divisible by 3, so -12/39 simplifies to -4/13. Thus, (-4/13)^(-3) becomes (13/-4)^3, which is (13^3) / (-4^3) = 2197 / -64.

Now, substitute the simplified exponential expressions back into the equation: (2197/-216) / y = 2197/-64. To solve for 'y', we first rewrite the division as multiplication by the reciprocal: (2197/-216) * (1/y) = 2197/-64. To isolate 'y', we can multiply both sides by the reciprocal of (2197/-216), which is -216/2197. Doing this, we get: 1/y = (2197/-64) * (-216/2197). The 2197 terms cancel out, leaving 1/y = -216/-64. Both numbers are negative, so the result is positive. Simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 8. This gives us 1/y = 27/8. To find 'y', we take the reciprocal of both sides, resulting in y = 8/27.

Therefore, the number by which we should divide (-6/13)^(-3) to obtain (-12/39)^(-3) is 8/27. This detailed walkthrough demonstrates the importance of simplifying expressions before solving equations and the power of algebraic manipulation in finding solutions.

Conclusion

These two problems illustrate how understanding the properties of exponents, especially negative exponents, and applying algebraic principles are crucial for solving mathematical challenges. By methodically simplifying expressions, setting up equations, and isolating the unknown variable, we can successfully navigate complex problems. The solutions not only provide numerical answers but also reinforce the value of a step-by-step approach in mathematics.

Mastering these concepts opens doors to more advanced mathematical topics and strengthens one's problem-solving skills. The ability to manipulate exponential expressions and solve equations is a fundamental skill that extends far beyond the classroom, finding applications in various fields of science, engineering, and finance. Therefore, a firm grasp of these principles is an invaluable asset in both academic and professional pursuits.