Solve ((3×8)^5)^6: Multiple Exponentiation Expression

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

The power of power rule says $ (a^m)^n = a^{m \times n} $. Think of it as: you're taking $ (3×8)^5 $ and multiplying it by itself 6 times, which requires multiplying the exponents!

Q: When do I add exponents then?

You add exponents when multiplying powers with the same base: $ a^m \times a^n = a^{m+n} $. But here we have nested powers, so we multiply the exponents instead.

Q: What's the difference between this and regular exponent multiplication?

Regular multiplication: $ x^3 \times x^4 = x^{3+4} = x^7 $ (add exponents). Power of power: $ (x^3)^4 = x^{3 \times 4} = x^{12} $ (multiply exponents).

Q: Should I calculate 3×8 first?

No need! The question asks for the expression, not the numerical value. Keep it as $ (3×8)^{30} $ or $ (3×8)^{5×6} $ depending on what form is requested.

Q: How can I remember this rule?

Think 'Power Tower': when you have powers stacked up like $ ((base)^{power1})^{power2} $, you multiply the powers to flatten the tower into $ base^{power1 \times power2} $.

Power of Power Rule with Multiple Exponents

Insert the corresponding expression:

$\left(\right.\left(3\times8\right)^5)^6=$

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Understand the problem

Insert the corresponding expression:

$\left(\right.\left(3\times8\right)^5)^6=$

Step-by-step solution

To solve the problem, we need to simplify the expression $\left(\left(3\times8\right)^5\right)^6$ .

We will utilize the "power of a power" rule in exponents, which states $(a^m)^n = a^{m \times n}$ . This rule tells us to multiply the exponents when raising a power to another power.

Step 1: Identify the expression to simplify: $\left(\left(3 \times 8\right)^5\right)^6$ .
Step 2: Apply the power of a power rule: This gives us $(3 \times 8)^{5 \times 6}$ .
Step 3: Multiply the exponents: $5 \times 6 = 30$ .

Therefore, the expression simplifies to $(3 \times 8)^{30}$ .

Upon comparing this result with the provided answer choices, the correct choice is:

$\left(3\times8\right)^{5\times6}$

This choice correctly applies the power of a power rule, thereby validating the solution as correct.

In conclusion, the simplified form of the expression is $(3 \times 8)^{30}$ , and the correct choice is option 4.

Final Answer

$\left(3\times8\right)^{5\times6}$

Key Points to Remember

Essential concepts to master this topic

Rule: When raising a power to another power, multiply the exponents
Technique: Apply $(a^m)^n = a^{m \times n}$ to get $(3×8)^{5 \times 6}$
Check: Final answer should be $(3×8)^{30}$ , not addition or subtraction ✓

Common Mistakes

Avoid these frequent errors

Adding or subtracting exponents instead of multiplying
Don't use $(3×8)^{5+6}$ or $(3×8)^{6-5}$ = wrong result! The power of power rule requires multiplication, not addition or subtraction. 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 of power rule says $(a^m)^n = a^{m \times n}$ . Think of it as: you're taking $(3×8)^5$ and multiplying it by itself 6 times, which requires multiplying the exponents!

When do I add exponents then?

You add exponents when multiplying powers with the same base: $a^m \times a^n = a^{m+n}$ . But here we have nested powers, so we multiply the exponents instead.

What's the difference between this and regular exponent multiplication?

Regular multiplication: $x^3 \times x^4 = x^{3+4} = x^7$ (add exponents).
Power of power: $(x^3)^4 = x^{3 \times 4} = x^{12}$ (multiply exponents).

Should I calculate 3×8 first?

No need! The question asks for the expression, not the numerical value. Keep it as $(3×8)^{30}$ or $(3×8)^{5×6}$ depending on what form is requested.

How can I remember this rule?

Think 'Power Tower': when you have powers stacked up like $((base)^{power1})^{power2}$ , you multiply the powers to flatten the tower into $base^{power1 \times power2}$ .

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