Simplify 6^-7 × 6^3: Negative and Positive Exponent Reduction

Q: Why do I add the exponents when multiplying?

The multiplication rule for exponents states $ a^m \times a^n = a^{m+n} $. This works because you're combining repeated multiplication: $ 6^2 \times 6^3 $ means (6×6) × (6×6×6) = 6 multiplied 5 times total = $ 6^5 $.

Q: What happens when I add a negative and positive exponent?

Just use regular addition rules! Negative + Positive = Subtract the smaller from the larger and keep the sign of the larger. So -7 + 3 = -4 because |-7| > |3|.

Q: How do I know if my negative exponent answer is right?

Convert to fractions to check! $ 6^{-4} = \frac{1}{6^4} $. Your original expression $ 6^{-7} \times 6^3 = \frac{1}{6^7} \times \frac{6^3}{1} = \frac{6^3}{6^7} = \frac{1}{6^4} $ ✓

Q: Can I multiply the bases together too?

No! Only add exponents when the bases are exactly the same. Don't multiply 6 × 6 = 36. The base stays as 6, and you only work with the exponents: -7 + 3 = -4.

Exponent Rules with Negative Powers

Reduce the following equation:

$6^{-7}\times6^3=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Simplify the following problem

00:03 According to the laws of exponents, the multiplication of powers with equal bases (A)

00:07 equals the same base raised to the sum of the exponents (N+M)

00:11 We will apply this formula to our exercise

00:15 We'll maintain the base and add the exponents together

00:19 This is the solution

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Reduce the following equation:

$6^{-7}\times6^3=$

Step-by-step solution

To simplify the expression $6^{-7} \times 6^3$ , we use the rule for multiplying powers with the same base, which states that we add the exponents together.

Given the expression:

$6^{-7} \times 6^3$

Step 1: Identify the base and the exponents involved. The base here is 6, with exponents $-7$ and $3$ .

Step 2: Apply the multiplication of powers rule:

$a^m \times a^n = a^{m+n}$

For our problem, $a = 6$ , $m = -7$ , and $n = 3$ . Therefore:

$6^{-7} \times 6^3 = 6^{-7 + 3}$

Step 3: Calculate the exponent:

$-7 + 3 = -4$

Therefore, the expression simplifies to:

$6^{-4}$

The solution to the equation is $6^{-4}$ .

Final Answer

$6^{-4}$

Key Points to Remember

Essential concepts to master this topic

Rule: When multiplying same bases, add the exponents together
Technique: $6^{-7} \times 6^3 = 6^{-7+3} = 6^{-4}$
Check: Verify by writing as fractions: $\frac{1}{6^7} \times \frac{6^3}{1} = \frac{1}{6^4}$ ✓

Common Mistakes

Avoid these frequent errors

Subtracting exponents when multiplying
Don't subtract -7 - 3 = -10 when multiplying! This gives $6^{-10}$ which is wrong. Multiplication means addition of exponents. Always add: -7 + 3 = -4 for the correct answer $6^{-4}$ .

Practice Quiz

Test your knowledge with interactive questions

$(3\times4\times5)^4=$

FAQ

Everything you need to know about this question

Why do I add the exponents when multiplying?

The multiplication rule for exponents states $a^m \times a^n = a^{m+n}$ . This works because you're combining repeated multiplication: $6^2 \times 6^3$ means (6×6) × (6×6×6) = 6 multiplied 5 times total = $6^5$ .

What happens when I add a negative and positive exponent?

Just use regular addition rules! Negative + Positive = Subtract the smaller from the larger and keep the sign of the larger. So -7 + 3 = -4 because |-7| > |3|.

How do I know if my negative exponent answer is right?

Convert to fractions to check! $6^{-4} = \frac{1}{6^4}$ . Your original expression $6^{-7} \times 6^3 = \frac{1}{6^7} \times \frac{6^3}{1} = \frac{6^3}{6^7} = \frac{1}{6^4}$ ✓

Can I multiply the bases together too?

No! Only add exponents when the bases are exactly the same. Don't multiply 6 × 6 = 36. The base stays as 6, and you only work with the exponents: -7 + 3 = -4.

What if I get confused about adding negative numbers?

Think of it as subtraction! -7 + 3 is the same as 3 - 7 = -4. Or use a number line: start at -7, move right 3 spaces, and you land on -4.

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