Simplify the Nested Expression: ((a²)³)^(1/4)

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

The power rule states $ (a^m)^n = a^{m \cdot n} $. You're not multiplying two separate powers - you're raising one power to another power, which requires multiplication of exponents.

Q: How do I handle multiple layers of parentheses?

Work from the inside out! First simplify $ (a^2)^3 = a^6 $, then apply the outer exponent: $ (a^6)^{1/4} = a^{6/4} = a^{3/2} $.

Q: Is a^1.5 the same as a^(3/2)?

Yes! $ \frac{3}{2} = 1\frac{1}{2} = 1.5 $. Both forms are correct - use whichever format the question asks for or is most convenient.

Q: What if the base has a negative sign?

Be careful with placement! $ (-a)^{3/2} $ means the negative is part of the base, while $ -a^{3/2} $ means you apply the negative after the exponent calculation.

Q: Can I use a calculator for fractional exponents?

Yes! Most calculators can handle fractional exponents. For $ a^{3/2} $, you can enter it as a^(3/2) or use the x^y function with 1.5 as the exponent.

Power Rules with Nested Exponents

$((a^2)^3)^{\frac{1}{4}}=$

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Understand the problem

$((a^2)^3)^{\frac{1}{4}}=$

Step-by-step solution

We use the power rule for exponents.

$(a^m)^n=a^{m\cdot n}$ We apply it to the problem:

$\big((a^2)^3\big)^{\frac{1}{4}}=(a^{2\cdot3})^{\frac{1}{4}}=a^{2\cdot3\cdot\frac{1}{4}}=a^{\frac{6}{4}}=a^{\frac{3}{2}}$ When we use the previously mentioned rule twice, the first time for the inner parentheses in the first stage and the second time for the remaining parentheses in the second stage, in the third stage we calculate the result of the multiplication in the exponent. While remembering that multiplying by a fraction is actually doubling the numerator of the fraction and, finally, in the last stage we simplify the fraction we obtained in the exponent.

Now remember that -

$\frac{3}{2}=1\frac{1}{2}=1.5$

Therefore, the correct answer is option a.

Final Answer

$a^{1.5}$

Key Points to Remember

Essential concepts to master this topic

Power Rule: When raising a power to a power, multiply the exponents
Technique: Apply $(a^m)^n = a^{m \cdot n}$ step by step from inside out
Check: Verify $a^{3/2} = a^{1.5}$ by converting fractions to decimals ✓

Common Mistakes

Avoid these frequent errors

Adding exponents instead of multiplying
Don't add the exponents 2 + 3 + 1/4 = 5.25! This gives a^5.25 instead of the correct answer. Adding only works when multiplying powers with the same base. Always multiply exponents when raising a power to another power.

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 rule states $(a^m)^n = a^{m \cdot n}$ . You're not multiplying two separate powers - you're raising one power to another power, which requires multiplication of exponents.

How do I handle multiple layers of parentheses?

Work from the inside out! First simplify $(a^2)^3 = a^6$ , then apply the outer exponent: $(a^6)^{1/4} = a^{6/4} = a^{3/2}$ .

Is a^1.5 the same as a^(3/2)?

Yes! $\frac{3}{2} = 1\frac{1}{2} = 1.5$ . Both forms are correct - use whichever format the question asks for or is most convenient.

What if the base has a negative sign?

Be careful with placement! $(-a)^{3/2}$ means the negative is part of the base, while $-a^{3/2}$ means you apply the negative after the exponent calculation.

Can I use a calculator for fractional exponents?

Yes! Most calculators can handle fractional exponents. For $a^{3/2}$ , you can enter it as a^(3/2) or use the x^y function with 1.5 as the exponent.

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Simplify the Nested Expression: ((a²)³)^(1/4)

Power Rules with Nested Exponents

❤️ Continue Your Math Journey!

Step-by-step video solution

Step-by-step written solution

Understand the problem

Step-by-step solution

Final Answer

Key Points to Remember

Common Mistakes

Practice Quiz

FAQ

Why do I multiply the exponents instead of adding them?

How do I handle multiple layers of parentheses?

Is a^1.5 the same as a^(3/2)?

What if the base has a negative sign?

Can I use a calculator for fractional exponents?

🌟 Unlock Your Math Potential

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