Simplify Powers of 15: Solving 15^4 × 15 × 15^3

Exponent Rules with Multiple Base Powers

Simplify the following equation:

154×15×153= 15^4\times15\times15^3=

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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 Any number raised to the power of 1 is always equal to itself
00:06 We will apply this formula to our exercise, and raise it to the power of 1
00:13 According to the laws of exponents, the multiplication of powers with an equal base (A)
00:17 equals the same base raised to the sum of the exponents (N+M)
00:21 We will apply this formula to our exercise
00:25 We'll maintain the base and add up the exponents
00:41 This is the solution

Step-by-step written solution

Follow each step carefully to understand the complete solution
1

Understand the problem

Simplify the following equation:

154×15×153= 15^4\times15\times15^3=

2

Step-by-step solution

To solve this problem, we'll employ the multiplication rule for exponents:

  • Step 1: Convert 1515 into exponential form: 15=15115 = 15^1.
  • Step 2: Apply the exponent rule to the expression 154×15×15315^4 \times 15 \times 15^3.
  • Step 3: Simplify by adding the exponents: 154×151×153=154+1+315^4 \times 15^1 \times 15^3 = 15^{4+1+3}.
  • Step 4: Perform the addition: 4+1+3=84 + 1 + 3 = 8.

Therefore, the simplified form of the expression is 15815^8.

The correct answer matches choice 3, which is: 15815^8.

3

Final Answer

158 15^8

Key Points to Remember

Essential concepts to master this topic
  • Rule: When multiplying same bases, add the exponents together
  • Technique: Convert 15 15 to 151 15^1 before adding: 4 + 1 + 3 = 8
  • Check: Count total factors: four 15's + one 15 + three 15's = eight 15's ✓

Common Mistakes

Avoid these frequent errors
  • Adding exponents without converting single bases
    Don't add 4 + 3 = 7 while ignoring the middle 15 = wrong answer 157 15^7 ! This misses one factor and gives an incomplete result. Always convert single bases like 15 to exponential form 151 15^1 first.

Practice Quiz

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

FAQ

Everything you need to know about this question

Why do I need to write 15 as 151 15^1 ?

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Because the exponent rule only works when all factors have exponents! Writing 15 as 151 15^1 shows it contributes 1 to the final exponent sum.

What if I just multiply 154×153 15^4 \times 15^3 and ignore the middle 15?

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You'll get 157 15^7 instead of the correct 158 15^8 ! That middle 15 is one more factor that must be included in your calculation.

Can I multiply the exponents instead of adding them?

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No! Multiplication of exponents happens with powers of powers like (154)3 (15^4)^3 . For multiplication of same bases, you always add the exponents.

How do I remember when to add vs multiply exponents?

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  • Same bases multiplied: am×an=am+n a^m \times a^n = a^{m+n} (ADD)
  • Power of a power: (am)n=am×n (a^m)^n = a^{m \times n} (MULTIPLY)

What's the fastest way to check my answer?

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Count the total number of 15's being multiplied: 154 15^4 means four 15's, 15 15 means one 15, and 153 15^3 means three 15's. Total: 4 + 1 + 3 = 8 factors!

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