Simplify (x³)⁴: Solving Compound Exponent Expression

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

When you have $ (x^3)^4 $, you're multiplying $ x^3 $ by itself 4 times. Each multiplication adds 3 to the exponent, so 3 + 3 + 3 + 3 = 12, which equals 3 × 4.

Q: What's the difference between $ x^3 \cdot x^4 $ and $ (x^3)^4 $ ?

Great question! $ x^3 \cdot x^4 = x^{3+4} = x^7 $ (add exponents), but $ (x^3)^4 = x^{3 \times 4} = x^{12} $ (multiply exponents). The parentheses and outer exponent tell you to use the power rule!

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

Multiplication: $ x^a \cdot x^b = x^{a+b} $ (add exponents)Power of a power: $ (x^a)^b = x^{a \times b} $ (multiply exponents)Look for parentheses with an outside exponent!

Q: Can I check my answer without expanding everything?

Yes! Try simple numbers: if $ x = 2 $, then $ (2^3)^4 = 8^4 = 4096 $ and $ 2^{12} = 4096 $. If both give the same result, you're correct!

Q: What if the base has a coefficient like $ (2x^3)^4 $ ?

Apply the power to everything inside: $ (2x^3)^4 = 2^4 \cdot (x^3)^4 = 16x^{12} $. The coefficient gets raised to the power too!

Power Rule with Compound Exponents

Insert the corresponding expression:

$\left(x^3\right)^4=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Simply

00:02 We'll use the power of power formula

00:04 Any number (A) to the power of (M) to the power of (N)

00:07 Equals the same number (A) to the power of the product of exponents (M*N)

00:10 We'll use this formula in our exercise

00:13 And we'll equate the numbers with the variables in the formula

00:25 We'll keep the base and multiply the exponents

00:39 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Insert the corresponding expression:

$\left(x^3\right)^4=$

Step-by-step solution

To simplify the expression $(x^3)^4$ , we'll follow these steps:

Step 1: Identify the expression: $(x^3)^4$ .
Step 2: Apply the formula for a power raised to another power.
Step 3: Calculate the product of the exponents.

Now, let's work through each step:

Step 1: We have the expression $(x^3)^4$ , which involves a power raised to another power.

Step 2: We apply the exponent rule $(a^m)^n = a^{m \cdot n}$ here with $a = x$ , $m = 3$ , and $n = 4$ .

Step 3: Multiply the exponents: $3 \times 4 = 12$ . This gives us a new exponent for the base $x$ .

Therefore, $(x^3)^4 = x^{12}$ .

Consequently, the correct answer choice is: $x^{12}$ from the options provided. The other options $x^6$ , $x^1$ , and $x^7$ do not reflect the correct application of the exponent multiplication rule.

Final Answer

$x^{12}$

Key Points to Remember

Essential concepts to master this topic

Power Rule: When raising a power to another power, multiply exponents
Technique: $(x^3)^4 = x^{3 \times 4} = x^{12}$
Check: Expand mentally: $x^3 \cdot x^3 \cdot x^3 \cdot x^3 = x^{12}$ ✓

Common Mistakes

Avoid these frequent errors

Adding exponents instead of multiplying
Don't add 3 + 4 = 7 to get $x^7$ ! This confuses the product rule with the power rule and gives completely wrong answers. Always multiply exponents when raising a power to another power: $(x^m)^n = x^{m \times n}$ .

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?

When you have $(x^3)^4$ , you're multiplying $x^3$ by itself 4 times. Each multiplication adds 3 to the exponent, so 3 + 3 + 3 + 3 = 12, which equals 3 × 4.

What's the difference between $x^3 \cdot x^4$ and $(x^3)^4$ ?

Great question! $x^3 \cdot x^4 = x^{3+4} = x^7$ (add exponents), but $(x^3)^4 = x^{3 \times 4} = x^{12}$ (multiply exponents). The parentheses and outer exponent tell you to use the power rule!

How can I remember when to multiply vs add exponents?

Multiplication: $x^a \cdot x^b = x^{a+b}$ (add exponents)
Power of a power: $(x^a)^b = x^{a \times b}$ (multiply exponents)
Look for parentheses with an outside exponent!

Can I check my answer without expanding everything?

Yes! Try simple numbers: if $x = 2$ , then $(2^3)^4 = 8^4 = 4096$ and $2^{12} = 4096$ . If both give the same result, you're correct!

What if the base has a coefficient like $(2x^3)^4$ ?

Apply the power to everything inside: $(2x^3)^4 = 2^4 \cdot (x^3)^4 = 16x^{12}$ . The coefficient gets raised to the power too!

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Simplify (x³)⁴: Solving Compound Exponent Expression

Power Rule with Compound 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?

What's the difference between $x^3 \cdot x^4$ and $(x^3)^4$ ?

How can I remember when to multiply vs add exponents?

Can I check my answer without expanding everything?

What if the base has a coefficient like $(2x^3)^4$ ?

🌟 Unlock Your Math Potential

More Questions

Applying Combined Exponents Rules

Simplify the Expression: y^9 ÷ y^3 Using Exponent Rules

Solve for 7^0: Calculating a Number Raised to Zero Power

Simplify the Power Fraction: x^6 divided by x^4

Simplify the Expression: b⁵/b² Using Exponent Rules

Simplify (x³)⁴: Solving Compound Exponent Expression

Power Rule with Compound 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?

What's the difference between x3⋅x4 x^3 \cdot x^4 x3⋅x4 and (x3)4 (x^3)^4 (x3)4?

How can I remember when to multiply vs add exponents?

Can I check my answer without expanding everything?

What if the base has a coefficient like (2x3)4 (2x^3)^4 (2x3)4?

🌟 Unlock Your Math Potential

More Questions

Applying Combined Exponents Rules

Simplify the Expression: y^9 ÷ y^3 Using Exponent Rules

Solve for 7^0: Calculating a Number Raised to Zero Power

Simplify the Power Fraction: x^6 divided by x^4

Simplify the Expression: b⁵/b² Using Exponent Rules

What's the difference between $x^3 \cdot x^4$ and $(x^3)^4$ ?

What if the base has a coefficient like $(2x^3)^4$ ?