Equivalent Expressions For √(36a⁸ / 225a²)

In this article, we will delve into simplifying the given expression ${\sqrt{\frac{36 a^8}{225 a^2}}}$ and identify which of the provided options are equivalent. This involves understanding the properties of square roots, exponents, and how to manipulate algebraic expressions. Our goal is to provide a comprehensive explanation that will help you not only solve this specific problem but also enhance your understanding of similar algebraic manipulations.

Breaking Down the Expression

To begin, let’s break down the expression step by step. The given expression is ${\sqrt{\frac{36 a^8}{225 a^2}}}$ which involves a square root of a fraction where both the numerator and the denominator contain coefficients and variables raised to powers. Simplifying such an expression requires a clear understanding of how to handle each component. First, we can separate the square root of the fraction into the fraction of the square roots:

${\sqrt{\frac{36 a^8}{225 a^2}} = \frac{\sqrt{36 a^8}}{\sqrt{225 a^2}}}$

This separation allows us to deal with the numerator and the denominator independently, simplifying the process. Next, we look at simplifying the square root of each part. For the numerator, we have ${\sqrt{36 a^8}}$. The square root of 36 is 6, since ${6^2 = 36}$. For the variable part, we recall that ${\sqrt{a^{2n}} = a^n}$, where ${n}$ is an integer. Thus, ${\sqrt{a^8} = a^4}$ because ${(a^4)^2 = a^8}$. Combining these, the square root of the numerator becomes:

${\sqrt{36 a^8} = 6a^4}$

Now, let’s simplify the denominator ${\sqrt{225 a^2}}$. The square root of 225 is 15, since ${15^2 = 225}$. The square root of ${a^2}$ is simply ${a}$, given the condition that ${a \neq 0}$. Therefore, the square root of the denominator is:

${\sqrt{225 a^2} = 15a}$

Having simplified both the numerator and the denominator, we can now rewrite the original expression as:

${\frac{\sqrt{36 a^8}}{\sqrt{225 a^2}} = \frac{6a^4}{15a}}$

Further Simplification

To further simplify the fraction ${\frac{6a^4}{15a}}$, we can reduce the numerical coefficients and the variable terms separately. The greatest common divisor (GCD) of 6 and 15 is 3. Dividing both the numerator and the denominator by 3, we get:

${\frac{6}{15} = \frac{6 \div 3}{15 \div 3} = \frac{2}{5}}$

For the variable terms, we have ${\frac{a^4}{a}}$. According to the quotient rule of exponents, ${\frac{a^m}{a^n} = a^{m-n}}$. Thus,

${\frac{a^4}{a} = a^{4-1} = a^3}$

Combining the simplified coefficients and variables, the expression becomes:

${\frac{6a^4}{15a} = \frac{2}{5} a^3}$

So, the simplified form of the original expression ${\sqrt{\frac{36 a^8}{225 a^2}}}$ is ${\frac{2}{5} a^3}$.

Identifying Equivalent Expressions

Now that we have the simplified form ${\frac{2}{5} a^3}$, we can compare it with the provided options (A, B, C, D, and E) to determine which expressions are equivalent. This requires us to manipulate each option, if necessary, to see if it matches our simplified expression. Let's assume we have the following options (as an example):

• A: ${\frac{2}{5} a^3}$
• B: ${\frac{6 a^4}{15 a}}$
• C: ${\frac{2 a^4}{5 a}}$
• D: ${\frac{36 a^8}{225 a^2}}$
• E: ${\frac{4 a^6}{10}}$

Analyzing Option A

Option A is ${\frac{2}{5} a^3}$. This expression is already in its simplest form and exactly matches our simplified expression. Therefore, option A is equivalent to ${\sqrt{\frac{36 a^8}{225 a^2}}}$ and should be checked.

Analyzing Option B

Option B is ${\frac{6 a^4}{15 a}}$. As we saw earlier in our simplification process, this expression is an intermediate step before we fully simplified the coefficients. To confirm, let’s simplify this expression: The fraction ${\frac{6}{15}}$ simplifies to ${\frac{2}{5}}$ by dividing both the numerator and the denominator by 3. For the variables, ${\frac{a^4}{a}}$ simplifies to ${a^3}$. Thus,

${\frac{6 a^4}{15 a} = \frac{2}{5} a^3}$

This matches our simplified expression, making option B equivalent.

Analyzing Option C

Option C is ${\frac{2 a^4}{5 a}}$. Simplifying this expression, we look at the variable part ${\frac{a^4}{a}}$, which, as we established, simplifies to ${a^3}$. Thus, the expression simplifies to:

${\frac{2 a^4}{5 a} = \frac{2}{5} a^3}$

This also matches our simplified expression, so option C is equivalent.

Analyzing Option D

Option D is ${\frac{36 a^8}{225 a^2}}$. This expression represents the radicand (the expression inside the square root) of our original problem. While it is a part of the original problem, it is not equivalent to the simplified form. To see this, remember our original simplification:

${\sqrt{\frac{36 a^8}{225 a^2}} = \frac{2}{5} a^3}$

The square root was a crucial part of the operation. Thus,

${\frac{36 a^8}{225 a^2} \neq \frac{2}{5} a^3}$

So, option D is not equivalent.

Analyzing Option E

Option E is ${\frac{4 a^6}{10}}$. To simplify this expression, we first simplify the fraction ${\frac{4}{10}}$, which reduces to ${\frac{2}{5}}$ by dividing both the numerator and the denominator by 2. So the expression becomes:

${\frac{4 a^6}{10} = \frac{2}{5} a^6}$

Comparing this to our simplified expression ${\frac{2}{5} a^3}$, we see that the exponents of ${a}$ are different. Therefore, option E is not equivalent.

Final Answer

In conclusion, the expressions equivalent to ${\sqrt{\frac{36 a^8}{225 a^2}}}$ are those that simplify to ${\frac{2}{5} a^3}$. Based on our analysis, Options A, B, and C are equivalent, while Options D and E are not. The process of simplification involves breaking down the expression, applying the rules of exponents and square roots, and then comparing the result with the given options. This exercise reinforces the importance of understanding algebraic manipulations and the properties of mathematical operations.

Additional Tips and Tricks

When faced with simplifying expressions involving radicals and exponents, remember these additional tips and tricks:

1. Always simplify inside the radical first: Before taking the square root (or any root), simplify the expression inside the radical. This often makes the entire process easier.
2. Look for perfect squares: Identifying perfect square factors (like 36, 225, or ${a^8}$) helps in simplifying square roots quickly.
3. Use exponent rules: Understand and apply the rules of exponents, such as the product rule, quotient rule, and power rule, to simplify variable terms.
4. Reduce fractions: Simplify numerical fractions by dividing both the numerator and the denominator by their greatest common divisor.
5. Double-check your work: After simplifying, always double-check your work by substituting a value for the variable (if possible) to ensure both the original and simplified expressions yield the same result.

By mastering these techniques, you can confidently tackle a wide range of algebraic simplification problems.