Simplify the Expression: (3^8 × a^8)/(x^8 × 5^8)

Q: Why can I combine terms under one exponent when they all have power 8?

This uses the reverse power rule! When you see $ a^n \times b^n $, it equals $ (a \times b)^n $. Since all four terms have exponent 8, we can factor out the 8th power.

Q: What if the exponents were different numbers?

Then you cannot combine them under a single power! The rule only works when all exponents are identical. Different exponents must be handled term by term.

Q: How do I know which form is most simplified?

The form $ \left(\frac{3a}{5x}\right)^8 $ is most simplified because it shows the base fraction clearly and reduces the visual complexity. It's easier to work with in further calculations.

Q: Can I multiply 3×a and 5×x in the final answer?

You can write it as $ \left(\frac{3a}{5x}\right)^8 $, but keep the multiplication signs when variables are involved. This makes it clear that a and x are separate factors, not combined into new variables.

Q: What if I expanded this back to check my work?

$ \left(\frac{3a}{5x}\right)^8 = \frac{(3a)^8}{(5x)^8} = \frac{3^8 \times a^8}{5^8 \times x^8} $This matches our original expression! ✓

Exponent Rules with Fraction Simplification

Insert the corresponding expression:

$\frac{3^8\times a^8}{x^8\times5^8}=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Simplify the following problem

00:03 According to the laws of exponents, a product that is raised to the power (N)

00:07 is equal to the product broken down into factors where each factor is raised to the power (N)

00:11 We will apply this formula to our exercise, in the reverse direction

00:18 We'll combine the factors into a multiplication operation within parentheses

00:28 According to the laws of exponents, a fraction raised to the power (N)

00:33 equals a fraction where the both numerator and denominator are raised to the power (N)

00:36 We will apply this formula to our exercise, in the reverse direction

00:42 We'll place the entire fraction inside of parentheses and raise it to the appropriate power

00:45 This is the solution

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Insert the corresponding expression:

$\frac{3^8\times a^8}{x^8\times5^8}=$

Step-by-step solution

To solve this problem, we'll rewrite the given fraction $\frac{3^8 \times a^8}{x^8 \times 5^8}$ using the rules of powers and exponents.

Step 1: Recognize the overall structure of the fraction is $\frac{m^8 \times n^8}{p^8 \times q^8}$ .
Step 2: Apply the rule of exponents to express it as a single power of the entire fraction: $\left(\frac{m \times n}{p \times q}\right)^8$ .
Step 3: Substitute the values: $\left(\frac{3 \times a}{x \times 5}\right)^8$ simplifies the expression effectively, given that all components are non-zero.

Thus, by applying the exponent rule directly to the entire fraction, we simplify to $\left(\frac{3 \times a}{x \times 5}\right)^8$ .

Final Answer

$\left(\frac{3\times a}{x\times5}\right)^8$

Key Points to Remember

Essential concepts to master this topic

Power Rule: When bases have same exponents, combine under single power
Technique: $\frac{a^n \times b^n}{c^n \times d^n} = \left(\frac{a \times b}{c \times d}\right)^n$
Check: Expand final answer to verify it equals original expression ✓

Common Mistakes

Avoid these frequent errors

Combining exponents incorrectly with different bases
Don't write $\frac{3^8 \times a^8}{x^8 \times 5^8}$ as $\frac{3a^8}{5x^8}$ = loses the power of 8 on constants! This changes the mathematical meaning completely. Always recognize when all terms share the same exponent and factor it out as a single power.

Practice Quiz

Test your knowledge with interactive questions

$112^0=\text{?}$

FAQ

Everything you need to know about this question

Why can I combine terms under one exponent when they all have power 8?

This uses the reverse power rule! When you see $a^n \times b^n$ , it equals $(a \times b)^n$ . Since all four terms have exponent 8, we can factor out the 8th power.

What if the exponents were different numbers?

Then you cannot combine them under a single power! The rule only works when all exponents are identical. Different exponents must be handled term by term.

How do I know which form is most simplified?

The form $\left(\frac{3a}{5x}\right)^8$ is most simplified because it shows the base fraction clearly and reduces the visual complexity. It's easier to work with in further calculations.

Can I multiply 3×a and 5×x in the final answer?

You can write it as $\left(\frac{3a}{5x}\right)^8$ , but keep the multiplication signs when variables are involved. This makes it clear that a and x are separate factors, not combined into new variables.

What if I expanded this back to check my work?

$\left(\frac{3a}{5x}\right)^8 = \frac{(3a)^8}{(5x)^8} = \frac{3^8 \times a^8}{5^8 \times x^8}$
This matches our original expression! ✓

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