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

00:00 Simplify the following problem

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

00:09 equals the same base raised to the sum of the exponents (N+M)

00:13 We'll apply this formula to our exercise

00:19 We'll maintain the base and add the exponents together

00:33 According to the exponent laws, any number with a negative exponent (-N)

00:36 equals its reciprocal raised to the opposite exponent (N)

00:39 We'll apply this formula to our exercise

00:42 We'll convert to the reciprocal and raise to the opposite exponent

00:48 This is 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

$112^0=\text{?}$

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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