Evaluate the Nested Expression: (2²)³ Using Exponent Rules

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

The power of a power rule says $ (a^m)^n = a^{m \times n} $. Think of it this way: $ (2^2)^3 $ means "multiply $ 2^2 $ by itself 3 times," which gives you 6 total factors of 2!

Q: When do I add exponents vs multiply them?

Add exponents when multiplying same bases: $ 2^3 \times 2^4 = 2^{3+4} = 2^7 $Multiply exponents when raising a power to a power: $ (2^3)^4 = 2^{3 \times 4} = 2^{12} $

Q: How can I remember which operation to use?

Look for parentheses! If you see $ (a^m)^n $ with parentheses around the first power, multiply the exponents. If there are no parentheses like $ a^m \times a^n $, then add the exponents.

Q: What if I calculated it step by step instead?

That works too! $ (2^2)^3 = (4)^3 = 4 \times 4 \times 4 = 64 $. Using the rule gives $ 2^{2 \times 3} = 2^6 = 64 $. Both methods should give the same answer!

Q: Does this rule work with negative or fractional exponents?

Yes! The power of a power rule works for all exponents. For example: $ (x^{-2})^3 = x^{-2 \times 3} = x^{-6} $ and $ (x^{1/2})^4 = x^{1/2 \times 4} = x^2 $.

Power of a Power with Multiplication Rule

Insert the corresponding expression:

$\left(2^2\right)^3=$

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Understand the problem

Insert the corresponding expression:

$\left(2^2\right)^3=$

Step-by-step solution

We are given the expression $(2^2)^3$ and need to simplify it using the laws of exponents and identify the corresponding expression among the choices.

To simplify the expression $(2^2)^3$ , we use the "power of a power" rule, which states that $(a^m)^n = a^{m \times n}$ .

Applying this rule to our expression, we have:

(2^2)^3 = 2^{2 \times 3}

Calculating the new exponent:

2 \times 3 = 6

Thus, the expression simplifies to:

2^6

Now, let's compare our result $2^6$ with the given choices:

Choice 1: $2^{2+3} = 2^5$ - Incorrect, as our expression evaluates to $2^6$ , not $2^5$ .
Choice 2: $2^{2-3} = 2^{-1}$ - Incorrect, as our expression evaluates to $2^6$ , not $2^{-1}$ .
Choice 3: $2^{\frac{2}{3}}$ - Incorrect, as our expression evaluates to $2^6$ , not a fractional exponent expression.
Choice 4: $2^{2 \times 3} = 2^6$ - Correct, as this matches our simplified expression.

Therefore, the correct choice is Choice 4: $2^{2 \times 3}$ .

Final Answer

$2^{2\times3}$

Key Points to Remember

Essential concepts to master this topic

Power Rule: When raising a power to a power, multiply the exponents
Technique: $(2^2)^3 = 2^{2 \times 3} = 2^6$ not addition
Check: Calculate both ways: $(2^2)^3 = 4^3 = 64$ and $2^6 = 64$ ✓

Common Mistakes

Avoid these frequent errors

Adding exponents instead of multiplying
Don't use $(2^2)^3 = 2^{2+3} = 2^5 = 32$ ! This confuses the power rule with the product rule and gives the wrong answer. Always multiply exponents when raising a power to a 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

Why do I multiply the exponents instead of adding them?

The power of a power rule says $(a^m)^n = a^{m \times n}$ . Think of it this way: $(2^2)^3$ means "multiply $2^2$ by itself 3 times," which gives you 6 total factors of 2!

When do I add exponents vs multiply them?

Add exponents when multiplying same bases: $2^3 \times 2^4 = 2^{3+4} = 2^7$
Multiply exponents when raising a power to a power: $(2^3)^4 = 2^{3 \times 4} = 2^{12}$

How can I remember which operation to use?

Look for parentheses! If you see $(a^m)^n$ with parentheses around the first power, multiply the exponents. If there are no parentheses like $a^m \times a^n$ , then add the exponents.

What if I calculated it step by step instead?

That works too! $(2^2)^3 = (4)^3 = 4 \times 4 \times 4 = 64$ . Using the rule gives $2^{2 \times 3} = 2^6 = 64$ . Both methods should give the same answer!

Does this rule work with negative or fractional exponents?

Yes! The power of a power rule works for all exponents. For example: $(x^{-2})^3 = x^{-2 \times 3} = x^{-6}$ and $(x^{1/2})^4 = x^{1/2 \times 4} = x^2$ .

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