Solve ((3×5)^-3)^-6: Nested Negative Exponents Challenge

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

The power of a power rule says $ (a^m)^n = a^{m \times n} $. We only add exponents when multiplying terms with the same base, like $ x^2 \times x^3 = x^5 $.

Q: What happens when I multiply two negative exponents?

Two negatives make a positive! So $ -3 \times -6 = +18 $. This is why $ ((3\times5)^{-3})^{-6} $ becomes $ (3\times5)^{18} $ with a positive exponent.

Q: How do I remember which exponent rule to use?

Look at the structure: If you see $ (something^{exponent})^{another\_exponent} $, use the power rule and multiply. If you see $ something^{exponent} \times something^{exponent} $, use the product rule and add.

Q: Can I simplify 3×5 first before applying the exponents?

You could calculate $ 3 \times 5 = 15 $ first, but the question asks for the expression format. The key is applying the power rule correctly regardless of whether you simplify the base.

Q: What if the answer choices don't show the multiplication sign?

Look for the choice that shows $ -3 \times -6 $ in the exponent, even if written differently. Some might use parentheses like $ (-3)(-6) $ or implicit multiplication.

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:

$\left(\left(3\times5\right)^{-3}\right)^{-6}=$

2

Step-by-step solution

To simplify the expression $\left(\left(3\times5\right)^{-3}\right)^{-6}$ , we apply exponent rules, specifically the power of a power rule.

Here's a step-by-step solution:

Step 1: Identify the structure of the expression: we have an outer exponent $-6$ and an inner exponent $-3$ applied to the base $(3 \times 5)$ .
Step 2: Apply the power of a power rule, which states $(a^m)^n = a^{m \cdot n}$ . This combines the exponents being multiplied.
Step 3: Calculate the multiplication of the exponents: $-3 \times -6$ . This yields $18$ since multiplying two negative numbers results in a positive number.
Step 4: Substitute back into the expression: $\left(3 \times 5\right)^{18}$ .

Thus, the transformation of the original expression results in the new expression:

$\left(3\times5\right)^{-3\times-6}$

Comparing with the provided choices, we see:

Choice 1: $\left(3\times5\right)^{-3\times-6}$ - This matches our solution.
Choices 2, 3, and 4 do not match the derived steps based on multiplying exponents.

Thus, the correct answer is choice 1: $\left(3\times5\right)^{-3\times-6}$ .

3

Final Answer

$\left(3\times5\right)^{-3\times-6}$

Key Points to Remember

Essential concepts to master this topic

Power Rule: When raising a power to a power, multiply the exponents
Technique: $(a^m)^n = a^{m \times n}$ so $-3 \times -6 = 18$
Check: Final result should be $(3\times5)^{18}$ with positive exponent ✓

Common Mistakes

Avoid these frequent errors

Adding exponents instead of multiplying
Don't add exponents like -3 + (-6) = -9! This confuses the power rule with the product rule and gives negative exponents when you should get positive ones. Always multiply exponents when raising a power to a power: (-3) × (-6) = 18.

Practice Quiz

Test your knowledge with interactive questions

$112^0=\text{?}$

FAQ

Everything you need to know about this question

Why do we multiply the exponents instead of adding them?

+

The power of a power rule says $(a^m)^n = a^{m \times n}$ . We only add exponents when multiplying terms with the same base, like $x^2 \times x^3 = x^5$ .

What happens when I multiply two negative exponents?

+

Two negatives make a positive! So $-3 \times -6 = +18$ . This is why $((3\times5)^{-3})^{-6}$ becomes $(3\times5)^{18}$ with a positive exponent.

How do I remember which exponent rule to use?

+

Look at the structure: If you see $(something^{exponent})^{another\_exponent}$ , use the power rule and multiply. If you see $something^{exponent} \times something^{exponent}$ , use the product rule and add.

Can I simplify 3×5 first before applying the exponents?

+

You could calculate $3 \times 5 = 15$ first, but the question asks for the expression format. The key is applying the power rule correctly regardless of whether you simplify the base.

What if the answer choices don't show the multiplication sign?

+

Look for the choice that shows $-3 \times -6$ in the exponent, even if written differently. Some might use parentheses like $(-3)(-6)$ or implicit multiplication.

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Solve ((3×5)^-3)^-6: Nested Negative Exponents Challenge

Power Rules with Negative Exponents

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