Simplify the Nested Exponent Expression: ((x×b)⁵)⁸

Q: When do I multiply exponents vs. add them?

Multiply exponents when raising a power to another power: $ (a^m)^n = a^{m \times n} $. Add exponents when multiplying powers with the same base: $ a^m \times a^n = a^{m+n} $.

Q: Why can't I just add 5 + 8 = 13?

Because you're raising an entire power to another power! Think of it as: $ ((x \times b)^5)^8 $ means multiplying $ (x \times b)^5 $ by itself 8 times, which requires multiplying the exponents.

Q: What if the base was different, like x and b separately?

If you had $ (x^5 \times b^5)^8 $, you'd apply the power to each factor: $ x^{5 \times 8} \times b^{5 \times 8} = x^{40} \times b^{40} $. But here, $ (x \times b) $ is treated as one base.

Q: How do I remember which operation to use with exponents?

Use this memory trick: "Same base multiplying? Add the powers. Power raised to power? Multiply them up!" Practice with simple numbers first, like $ (2^3)^4 = 2^{12} $.

Q: Can I simplify this expression further?

Not without knowing specific values for x and b. $ (x \times b)^{40} $ is the simplest form using exponent rules. You could expand it as $ x^{40} \times b^{40} $, but that's usually not necessary.

Step-by-step video solution

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Step-by-step written solution

Follow each step carefully to understand the complete solution

1

Understand the problem

Insert the corresponding expression:

$\left(\left(x\times b\right)^5\right)^8=$

2

Step-by-step solution

To solve this problem, we'll follow these steps:

Step 1: Identify the mathematical problem and the rule needed.
Step 2: Apply the power of a power rule to simplify the expression.
Step 3: Evaluate our simplification against the provided answer choices.

Now, let's work through each step:

Step 1: The problem involves simplifying the expression $\left(\left(x \times b\right)^5\right)^8$ , which is a power of a power. Our goal is to simplify it to a single power.

Step 2: According to the power of a power rule, $(a^m)^n = a^{m \times n}$ . In this case, the base is $(x \times b)$ raised to the 5th power and that entire expression is further raised to the 8th power.

Applying the power of a power rule gives us:

$\left(\left(x \times b\right)^5\right)^8 = \left(x \times b\right)^{5 \times 8} = \left(x \times b\right)^{40}$

Step 3: Compare this to the provided answer choices:

Choice 1: $\left(x \times b\right)^{5+8}$ – This is incorrect as it adds the exponents instead of multiplying them.
Choice 2: $\left(x \times b\right)^{5 \times 8}$ – This is correct as it applies the rule correctly to yield $(x \times b)^{40}$ .
Choice 3: $\left(x \times b\right)^{\frac{8}{5}}$ – This is incorrect and irrelevant to the power of a power rule.
Choice 4: $\left(x \times b\right)^{5-8}$ – This is incorrect as it subtracts exponents, which is not applicable here.

Therefore, after careful analysis, the solution to the problem is the expression represented by Choice 2: $\left(x \times b\right)^{40}$ .

3

Final Answer

$\left(x\times b\right)^{5\times8}$

Key Points to Remember

Essential concepts to master this topic

Power Rule: When raising a power to another power, multiply the exponents
Technique: $(a^m)^n = a^{m \times n}$ , so $((x \times b)^5)^8 = (x \times b)^{5 \times 8}$
Check: Verify that 5 × 8 = 40, so the final answer is $(x \times b)^{40}$ ✓

Common Mistakes

Avoid these frequent errors

Adding exponents instead of multiplying them
Don't add the exponents like $(x \times b)^{5+8} = (x \times b)^{13}$ ! This confuses the power rule with the product rule and gives completely wrong results. Always multiply the exponents when raising a power to another power: $(a^m)^n = a^{m \times n}$ .

Practice Quiz

Test your knowledge with interactive questions

$112^0=\text{?}$

FAQ

Everything you need to know about this question

When do I multiply exponents vs. add them?

+

Multiply exponents when raising a power to another power: $(a^m)^n = a^{m \times n}$ . Add exponents when multiplying powers with the same base: $a^m \times a^n = a^{m+n}$ .

Why can't I just add 5 + 8 = 13?

+

Because you're raising an entire power to another power! Think of it as: $((x \times b)^5)^8$ means multiplying $(x \times b)^5$ by itself 8 times, which requires multiplying the exponents.

What if the base was different, like x and b separately?

+

If you had $(x^5 \times b^5)^8$ , you'd apply the power to each factor: $x^{5 \times 8} \times b^{5 \times 8} = x^{40} \times b^{40}$ . But here, $(x \times b)$ is treated as one base.

How do I remember which operation to use with exponents?

+

Use this memory trick: "Same base multiplying? Add the powers. Power raised to power? Multiply them up!" Practice with simple numbers first, like $(2^3)^4 = 2^{12}$ .

Can I simplify this expression further?

+

Not without knowing specific values for x and b. $(x \times b)^{40}$ is the simplest form using exponent rules. You could expand it as $x^{40} \times b^{40}$ , but that's usually not necessary.

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Simplify the Nested Exponent Expression: ((x×b)⁵)⁸

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