Simplify the Expression: 6^-5 × 6^2 Using Laws of Exponents

Q: Why do I add -5 and 2 instead of subtracting?

The exponent rule for multiplication is always add the exponents, even with negative numbers. Think of it as: $ (-5) + 2 = -3 $, just like regular addition!

Q: What does a negative exponent actually mean?

A negative exponent means "take the reciprocal". So $ 6^{-3} = \frac{1}{6^3} $. The negative flips the number to the denominator of a fraction.

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

Multiplication = Add exponents, Division = Subtract exponents. Since we're multiplying $ 6^{-5} \times 6^2 $, we add: -5 + 2 = -3.

Q: Is there a way to check if my final answer is correct?

Yes! Calculate $ \frac{1}{6^3} = \frac{1}{216} $ and also calculate $ 6^{-5} \times 6^2 $ step by step: $ \frac{1}{6^5} \times 6^2 = \frac{6^2}{6^5} = \frac{1}{6^3} $ ✓

Q: Why isn't the answer just 6³?

Because we have $ 6^{-3} $, not $ 6^3 $! The negative exponent makes all the difference - it means we need the reciprocal: $ \frac{1}{6^3} $.

Negative Exponents with Same Base Multiplication

Insert the corresponding expression:

$6^{-5}\times6^2=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:06 Let's simplify this problem together.

00:10 When we multiply exponents with the same base, like A,

00:15 we keep the base and add the exponents, N plus M.

00:20 So, let's apply this to our problem now.

00:25 Remember, maintain the base and add those exponents together.

00:39 For a negative exponent, like negative N,

00:43 it's the same as taking the reciprocal and using N.

00:48 Let's use this rule in our exercise.

00:52 We'll convert to the reciprocal and raise to the opposite exponent.

00:57 And there you have it, that's the solution!

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Insert the corresponding expression:

$6^{-5}\times6^2=$

Step-by-step solution

Let's solve the problem step-by-step:

Step 1: Identify the base and exponents involved.
Step 2: Apply the exponent multiplication rule to simplify.
Step 3: Further simplify the expression if necessary.

Now, we will work through each step:

Step 1: The problem gives us the expression $6^{-5} \times 6^2$ . We have a common base, which is 6.

Step 2: Using the rule for multiplying exponents with the same base, we add the exponents. Thus, $6^{-5} \times 6^2 = 6^{-5+2} = 6^{-3}$ .

Step 3: Simplifying further, since a negative exponent means the reciprocal, we have:
$6^{-3} = \frac{1}{6^3}$ .

Therefore, the solution to the problem is: $\frac{1}{6^3}$ .

Final Answer

$\frac{1}{6^3}$

Key Points to Remember

Essential concepts to master this topic

Rule: When multiplying same bases, add the exponents together
Technique: Calculate $6^{-5} \times 6^2 = 6^{-5+2} = 6^{-3}$
Check: Convert negative exponent: $6^{-3} = \frac{1}{6^3} = \frac{1}{216}$ ✓

Common Mistakes

Avoid these frequent errors

Subtracting exponents instead of adding them
Don't calculate $6^{-5} \times 6^2$ as $6^{-5-2} = 6^{-7}$ = wrong answer! This confuses multiplication with division rules. Always add exponents when multiplying same bases, regardless of negative signs.

Practice Quiz

Test your knowledge with interactive questions

Simplify the following equation:

$$ $4^5\times4^5=$

FAQ

Everything you need to know about this question

Why do I add -5 and 2 instead of subtracting?

The exponent rule for multiplication is always add the exponents, even with negative numbers. Think of it as: $(-5) + 2 = -3$ , just like regular addition!

What does a negative exponent actually mean?

A negative exponent means "take the reciprocal". So $6^{-3} = \frac{1}{6^3}$ . The negative flips the number to the denominator of a fraction.

How can I remember when to add vs subtract exponents?

Multiplication = Add exponents, Division = Subtract exponents. Since we're multiplying $6^{-5} \times 6^2$ , we add: -5 + 2 = -3.

Is there a way to check if my final answer is correct?

Yes! Calculate $\frac{1}{6^3} = \frac{1}{216}$ and also calculate $6^{-5} \times 6^2$ step by step: $\frac{1}{6^5} \times 6^2 = \frac{6^2}{6^5} = \frac{1}{6^3}$ ✓

Why isn't the answer just 6³?

Because we have $6^{-3}$ , not $6^3$ ! The negative exponent makes all the difference - it means we need the reciprocal: $\frac{1}{6^3}$ .

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