Solve (4·7)^9 + 2^7/2^4 + (8^2)^5: Complex Exponent Calculation

Q: When do I add exponents versus subtract them?

Add exponents when multiplying: $ a^m \cdot a^n = a^{m+n} $Subtract exponents when dividing: $ \frac{a^m}{a^n} = a^{m-n} $Remember: multiplication adds, division subtracts!

Q: What's the difference between $ (8^2)^5 $ and $ 8^2 \cdot 8^5 $ ?

$ (8^2)^5 = 8^{2 \times 5} = 8^{10} $ (power raised to a power = multiply exponents)$ 8^2 \cdot 8^5 = 8^{2+5} = 8^7 $ (same base multiplication = add exponents)These give completely different results!

Q: Why doesn't $ (4 \cdot 7)^9 $ become $ 4^9 \cdot 7^9 $ ?

Actually, it could become $ 4^9 \cdot 7^9 $ using the rule $ (ab)^n = a^n b^n $, but that makes the problem much harder!It's easier to multiply inside first: $ (4 \cdot 7)^9 = 28^9 $

Q: Can I use a calculator for these large exponents?

For this problem, you don't need to calculate the actual values! The question asks for the simplified form.$ 28^9 + 2^3 + 8^{10} $ is the final answer, not the huge number you'd get from calculating each term.

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

Same base, multiplying? Add exponents: $ a^m \cdot a^n = a^{m+n} $Same base, dividing? Subtract exponents: $ \frac{a^m}{a^n} = a^{m-n} $Power raised to power? Multiply exponents: $ (a^m)^n = a^{m \cdot n} $

Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Solve the following problem

00:03 When there's a power over a product of numbers, all terms are raised to that power

00:08 When dividing powers with equal bases

00:13 The power of the result equals the difference of the powers

00:19 Let's use this formula in our exercise

00:23 When there's a power of a power, the resulting power is the product of the powers

00:29 Let's use this formula in our exercise

00:33 Let's calculate all the powers

00:43 This is the solution

Step-by-step written solution

Follow each step carefully to understand the complete solution

1

Understand the problem

$(4\cdot7)^9+\frac{2^7}{2^4}+(8^2)^5=$

2

Step-by-step solution

In order to solve the problem we must use two power laws, as shown below:

A. Power property for terms with identical bases:

$\frac{a^m}{a^n}=a^{m-n}$ B. Power property for an exponent raised to another exponent:

$(a^m)^n=a^{m\cdot n}$

We will apply these two power laws to the problem in two steps:

Let's start by applying the power law specified in A to the second term from the left in the given problem:

$\frac{2^7}{2^4}=2^{7-4}=2^3$ In the first step we apply the power law specified in A and then proceed to simplify the resulting expression,

We then advance to the next step and apply the power law specified in B to the third term from the left in the given problem :

$(8^2)^5=8^{2\cdot5}=8^{10}$ In the first stage we apply the power law specified in B and then proceed to simplify the resulting expression,

Let's summarize the two steps listed above to solve the general problem:

$(4\cdot7)^9+\frac{2^7}{2^4}+(8^2)^5= (4\cdot7)^9+2^3+8^{10}$ In the final step, we calculate the result of multiplying the terms within the parentheses in the first term from the left:

$(4\cdot7)^9+2^3+8^{10}=28^9+2^3+8^{10}$ Therefore, the correct answer is option c.

3

Final Answer

$28^9+2^3+8^{10}$

Key Points to Remember

Essential concepts to master this topic

Division Rule: $\frac{a^m}{a^n} = a^{m-n}$ when bases are identical
Power Rule: $(a^m)^n = a^{m \cdot n}$ like $(8^2)^5 = 8^{10}$
Verification: Check each step: $2^7/2^4 = 2^3 = 8$ and $(8^2)^5 = 8^{10}$ ✓

Common Mistakes

Avoid these frequent errors

Adding exponents when dividing powers
Don't add exponents like $\frac{2^7}{2^4} = 2^{11}$ = wrong answer! This confuses multiplication rules with division rules. Always subtract exponents when dividing: $\frac{2^7}{2^4} = 2^{7-4} = 2^3$ .

Practice Quiz

Test your knowledge with interactive questions

$112^0=\text{?}$

FAQ

Everything you need to know about this question

When do I add exponents versus subtract them?

+

Add exponents when multiplying: $a^m \cdot a^n = a^{m+n}$

Subtract exponents when dividing: $\frac{a^m}{a^n} = a^{m-n}$

Remember: multiplication adds, division subtracts!

What's the difference between $(8^2)^5$ and $8^2 \cdot 8^5$ ?

+

$(8^2)^5 = 8^{2 \times 5} = 8^{10}$ (power raised to a power = multiply exponents)

$8^2 \cdot 8^5 = 8^{2+5} = 8^7$ (same base multiplication = add exponents)

These give completely different results!

Why doesn't $(4 \cdot 7)^9$ become $4^9 \cdot 7^9$ ?

+

Actually, it could become $4^9 \cdot 7^9$ using the rule $(ab)^n = a^n b^n$ , but that makes the problem much harder!

It's easier to multiply inside first: $(4 \cdot 7)^9 = 28^9$

Can I use a calculator for these large exponents?

+

For this problem, you don't need to calculate the actual values! The question asks for the simplified form.

$28^9 + 2^3 + 8^{10}$ is the final answer, not the huge number you'd get from calculating each term.

How do I remember which exponent rule to use?

+

Same base, multiplying? Add exponents: $a^m \cdot a^n = a^{m+n}$
Same base, dividing? Subtract exponents: $\frac{a^m}{a^n} = a^{m-n}$
Power raised to power? Multiply exponents: $(a^m)^n = a^{m \cdot n}$

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Solve (4·7)^9 + 2^7/2^4 + (8^2)^5: Complex Exponent Calculation

Exponent Rules with Multiple Operations

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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

When do I add exponents versus subtract them?

What's the difference between (82)5 (8^2)^5 (82)5 and 82⋅85 8^2 \cdot 8^5 82⋅85?

Why doesn't (4⋅7)9 (4 \cdot 7)^9 (4⋅7)9 become 49⋅79 4^9 \cdot 7^9 49⋅79?

Can I use a calculator for these large exponents?

How do I remember which exponent rule to use?

🌟 Unlock Your Math Potential

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What's the difference between $(8^2)^5$ and $8^2 \cdot 8^5$ ?

Why doesn't $(4 \cdot 7)^9$ become $4^9 \cdot 7^9$ ?