Simplify the Expression: 9^-3 × 9^-5 × 9^-2 Using Exponent Laws

Exponent Laws with Negative Powers

Reduce the following equation:

9−3×9−5×9−2= 9^{-3}\times9^{-5}\times9^{-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 power laws, the multiplication of powers with equal bases (A)
00:06 equals the same base raised to the sum of the exponents (N+M)
00:12 We will apply this formula to our exercise
00:15 Note that we are adding together negative factors
00:33 A positive x A negative is always negative, therefore we subtract as follows
00:45 This is the solution

Step-by-step written solution

Follow each step carefully to understand the complete solution
1

Understand the problem

Reduce the following equation:

9−3×9−5×9−2= 9^{-3}\times9^{-5}\times9^{-2}=

2

Step-by-step solution

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

  • Step 1: Identify the common base.
  • Step 2: Apply the multiplication of powers rule.
  • Step 3: Simplify by adding the exponents.

Now, let's work through each step:
Step 1: The base in every term of the expression 9−3×9−5×9−2 9^{-3} \times 9^{-5} \times 9^{-2} is 9.
Step 2: Use the exponent multiplication rule am×an=am+n a^m \times a^n = a^{m+n} .
Step 3: Add the exponents: −3+(−5)+(−2)=−10-3 + (-5) + (-2) = -10.
The expression simplifies to 9−10 9^{-10} .

Therefore, the solution to the problem is 9−10 9^{-10} .

3

Final Answer

9−10 9^{-10}

Key Points to Remember

Essential concepts to master this topic
  • Rule: When multiplying powers with same base, add the exponents
  • Technique: 9−3×9−5×9−2 9^{-3} \times 9^{-5} \times 9^{-2} becomes 9−3+(−5)+(−2) 9^{-3+(-5)+(-2)}
  • Check: Count exponents: three negative values should give one negative result ✓

Common Mistakes

Avoid these frequent errors
  • Multiplying the exponents instead of adding them
    Don't multiply -3 × -5 × -2 = -30 to get 9−30 9^{-30} ! This confuses multiplication of powers with powers of powers. Always add exponents when multiplying same bases: -3 + (-5) + (-2) = -10.

Practice Quiz

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\( 112^0=\text{?} \)

FAQ

Everything you need to know about this question

Why do I add negative exponents instead of subtracting them?

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Adding negative numbers is the same as subtracting! When you see -3 + (-5), you're really doing -3 - 5 = -8. The key is to always add exponents when multiplying powers with the same base.

What's the difference between 9−10 9^{-10} and 910 9^{10} ?

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9−10 9^{-10} means 1 divided by 910 9^{10} , which is a very small positive number. 910 9^{10} is a huge positive number. The negative exponent flips the fraction!

Can I change the order of multiplication?

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Yes! Multiplication is commutative, so 9−3×9−5×9−2 9^{-3} \times 9^{-5} \times 9^{-2} equals 9−2×9−3×9−5 9^{-2} \times 9^{-3} \times 9^{-5} . You'll still add the same exponents and get 9−10 9^{-10} .

How do I remember when to add versus multiply exponents?

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Same base being multiplied = ADD exponents. Power raised to a power = MULTIPLY exponents. For example: (9−3)2=9−6 (9^{-3})^2 = 9^{-6} because -3 × 2 = -6.

What if I forget the exponent rule during a test?

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Try a simple example first! 23×22=8×4=32=25 2^3 \times 2^2 = 8 \times 4 = 32 = 2^5 . Notice that 3 + 2 = 5, not 3 × 2 = 6. This helps you remember to add the exponents!

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