Solve the Nested Expression: ((b×6)^5)^2 Using Exponent Rules

Q: Why do I multiply the exponents instead of adding them?

The power of a power rule says $ (x^m)^n = x^{m \times n} $. You only add exponents when multiplying same bases: $ x^2 \times x^3 = x^5 $. Here we're raising one power to another power!

Q: What if there are multiple sets of parentheses like this?

Work from the innermost parentheses outward! Apply the power rule step by step. For example: $ (((x^2)^3)^4) = ((x^6)^4) = x^{24} $

Q: Does the order of operations matter here?

Yes! Exponents are evaluated from right to left, so $ ((b \times 6)^5)^2 $ means raise $ (b \times 6)^5 $ to the 2nd power, not the other way around.

Q: Can I distribute the outer exponent to each factor inside?

Not directly in this case! You must first apply the power of a power rule to get $ (b \times 6)^{10} $. Then you could distribute: $ b^{10} \times 6^{10} $, but that's not one of the answer choices.

Q: How do I remember when to multiply vs add exponents?

Multiply: Power to a power $ (x^m)^n = x^{mn} $Add: Same base multiplication $ x^m \times x^n = x^{m+n} $Think: "Power to power = multiply!"

Exponent Rules with Nested Expressions

Insert the corresponding expression:

$\left(\left(b\times6\right)^5\right)^2=$

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Understand the problem

Insert the corresponding expression:

$\left(\left(b\times6\right)^5\right)^2=$

Step-by-step solution

To solve the problem, we need to simplify the expression $\left(\left(b \times 6\right)^5\right)^2$ .

We will apply the power of a power rule in exponents, which states:

For an expression $(x^m)^n$ , it simplifies to $x^{m \times n}$ .

Applying this rule to our expression:

$\left(\left(b \times 6\right)^5\right)^2 = \left(b \times 6\right)^{5 \times 2}$

Calculating the new exponent:

$5 \times 2 = 10$

Therefore, the simplified expression is:

$\left(b \times 6\right)^{10}$

We will now compare this to the given multiple-choice answers:

Choice 1: $\left(b\times6\right)^3$ - Incorrect
Choice 2: $\left(b\times6\right)^{10}$ - Correct
Choice 3: $\left(b\times6\right)^7$ - Incorrect
Choice 4: $\left(b\times6\right)^{\frac{2}{5}}$ - Incorrect

In conclusion, the correct answer is $\left(b\times6\right)^{10}$ , which matches Choice 2.

Final Answer

$\left(b\times6\right)^{10}$

Key Points to Remember

Essential concepts to master this topic

Power Rule: When raising a power to a power, multiply the exponents
Technique: $((b \times 6)^5)^2 = (b \times 6)^{5 \times 2} = (b \times 6)^{10}$
Check: Count parentheses and apply rules in correct order to get $(b \times 6)^{10}$ ✓

Common Mistakes

Avoid these frequent errors

Adding exponents instead of multiplying
Don't add 5 + 2 = 7 when you see $((b \times 6)^5)^2$ = $(b \times 6)^7$ ! This confuses the product rule with the power rule. Always multiply exponents when raising a power to a power: 5 × 2 = 10.

Practice Quiz

Test your knowledge with interactive questions

$112^0=\text{?}$

FAQ

Everything you need to know about this question

Why do I multiply the exponents instead of adding them?

The power of a power rule says $(x^m)^n = x^{m \times n}$ . You only add exponents when multiplying same bases: $x^2 \times x^3 = x^5$ . Here we're raising one power to another power!

What if there are multiple sets of parentheses like this?

Work from the innermost parentheses outward! Apply the power rule step by step. For example: $(((x^2)^3)^4) = ((x^6)^4) = x^{24}$

Does the order of operations matter here?

Yes! Exponents are evaluated from right to left, so $((b \times 6)^5)^2$ means raise $(b \times 6)^5$ to the 2nd power, not the other way around.

Can I distribute the outer exponent to each factor inside?

Not directly in this case! You must first apply the power of a power rule to get $(b \times 6)^{10}$ . Then you could distribute: $b^{10} \times 6^{10}$ , but that's not one of the answer choices.

How do I remember when to multiply vs add exponents?

Multiply: Power to a power $(x^m)^n = x^{mn}$
Add: Same base multiplication $x^m \times x^n = x^{m+n}$

Think: "Power to power = multiply!"

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