Simplify 4⁷ × 5³ × 4² × 5⁴: Exponent Multiplication Problem

Q: Why can't I just add all the exponents together?

You can only add exponents when the bases are the same! In this problem, 4 and 5 are different bases, so $ 4^7 \times 5^3 $ stays separate. Only combine $ 4^7 \times 4^2 $ and $ 5^3 \times 5^4 $.

Q: Do I need to calculate the final numerical answer?

Not unless specifically asked! Leaving the answer as $ 4^9 \times 5^7 $ is the simplified form and is usually preferred. Computing $ 4^9 \times 5^7 $ would give a very large number!

Q: How do I remember which exponents to add?

Think of it like sorting! Put all the 4's together: $ 4^7 \times 4^2 $, and all the 5's together: $ 5^3 \times 5^4 $. Then add the exponents within each group.

Q: What if there were three different bases?

Same process! Group each base separately. For example, $ 2^3 \times 3^4 \times 2^5 \times 3^2 \times 7^1 = 2^8 \times 3^6 \times 7^1 $. Each base gets its own group.

Exponent Multiplication with Same Bases

Simplify the following equation:

$4^7\times5^3\times4^2\times5^4=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Simply

00:04 We'll use the substitution law and arrange the equal bases together

00:14 We'll use the formula for multiplying powers with powers

00:16 Any number (A) to the power of (M) multiplied by the same number (A) to the power of (N)

00:19 Equals the same number (A) to the power of the sum of exponents (M+N)

00:23 We'll use this formula in our exercise

00:29 And equate the numbers to the variables in the formula

00:40 We'll keep the base and combine the exponents

01:15 We'll use the same method for the second base

01:32 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

Simplify the following equation:

$4^7\times5^3\times4^2\times5^4=$

Step-by-step solution

To solve this problem, we'll follow these steps:

Step 1: Identify and group the terms with the same base.
Step 2: Apply the laws of exponents to simplify by adding the exponents of each base.
Step 3: Write the simplified form.

Let's work through each step:

Step 1: We are given that $4^7 \times 5^3 \times 4^2 \times 5^4$ .

Step 2: First, group the terms with the same base:

$4^7 \times 4^2$ and $5^3 \times 5^4$ .

Step 3: Use the law of exponents, which states $a^m \times a^n = a^{m+n}$ .

For the base 4: $4^7 \times 4^2 = 4^{7+2} = 4^9$ .

For the base 5: $5^3 \times 5^4 = 5^{3+4} = 5^7$ .

Therefore, the simplified form of the expression is $4^9 \times 5^7$ .

Final Answer

$4^9\times5^7$

Key Points to Remember

Essential concepts to master this topic

Law of Exponents: When multiplying same bases, add their exponents
Technique: Group terms: $4^7 \times 4^2 = 4^{7+2} = 4^9$
Check: Verify each base separately: 4 terms give $4^9$ , 5 terms give $5^7$ ✓

Common Mistakes

Avoid these frequent errors

Adding exponents across different bases
Don't add 7+3+2+4=16 to get one exponent = completely wrong answer! This mixes different bases together incorrectly. Always group terms with the same base first, then add exponents within each group separately.

Practice Quiz

Test your knowledge with interactive questions

$112^0=\text{?}$

FAQ

Everything you need to know about this question

Why can't I just add all the exponents together?

You can only add exponents when the bases are the same! In this problem, 4 and 5 are different bases, so $4^7 \times 5^3$ stays separate. Only combine $4^7 \times 4^2$ and $5^3 \times 5^4$ .

What if the bases were all the same number?

Great question! If all terms had the same base (like all 4's), then you would add all the exponents together. For example, $4^7 \times 4^3 \times 4^2 \times 4^4 = 4^{16}$ .

Do I need to calculate the final numerical answer?

Not unless specifically asked! Leaving the answer as $4^9 \times 5^7$ is the simplified form and is usually preferred. Computing $4^9 \times 5^7$ would give a very large number!

How do I remember which exponents to add?

Think of it like sorting! Put all the 4's together: $4^7 \times 4^2$ , and all the 5's together: $5^3 \times 5^4$ . Then add the exponents within each group.

What if there were three different bases?

Same process! Group each base separately. For example, $2^3 \times 3^4 \times 2^5 \times 3^2 \times 7^1 = 2^8 \times 3^6 \times 7^1$ . Each base gets its own group.

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