Solve: 5³ + 5⁻³ × 5³ Expression with Mixed Exponents

Exponent Laws with Negative Powers

53+5−3⋅53=? 5^3+5^{-3}\cdot5^3=\text{?}

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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 When multiplying powers with equal bases
00:08 The power of the result equals the sum of the powers
00:12 We'll apply this formula to our exercise, and proceed to add the powers together
00:23 According to the laws of powers, any number raised to the power of 0 equals 1
00:27 As long as the number is not 0
00:30 We'll apply this formula to our exercise
00:33 This is the solution

Step-by-step written solution

Follow each step carefully to understand the complete solution
1

Understand the problem

53+5−3⋅53=? 5^3+5^{-3}\cdot5^3=\text{?}

2

Step-by-step solution

We'll use the power rule for multiplication between terms with identical bases:

am⋅an=am+n a^m\cdot a^n=a^{m+n} and we'll simplify the second term on the left in the equation using it:
53+5−3⋅53=53+5−3+3=53+50=53+1 5^3+5^{-3}\cdot5^3=5^3+5^{-3+3}=5^3+5^0=5^3+1 where in the first stage we applied the mentioned rule to the second term on the left, then we simplified the expression with the exponent, and in the final stage we used the fact that any number raised to the power of 0 equals 1,

We didn't touch the first term of course since it was already simplified,

Therefore the correct answer is answer C.

3

Final Answer

53+1 5^3+1

Key Points to Remember

Essential concepts to master this topic
  • Rule: When multiplying same bases, add the exponents: am⋅an=am+n a^m \cdot a^n = a^{m+n}
  • Technique: Simplify 5−3⋅53=5−3+3=50=1 5^{-3} \cdot 5^3 = 5^{-3+3} = 5^0 = 1 first
  • Check: Verify that 53+1=125+1=126 5^3 + 1 = 125 + 1 = 126 ✓

Common Mistakes

Avoid these frequent errors
  • Adding exponents incorrectly or ignoring the multiplication rule
    Don't compute 53+5−3⋅53 5^3 + 5^{-3} \cdot 5^3 as 53+5−3+53=2⋅53+5−3 5^3 + 5^{-3} + 5^3 = 2 \cdot 5^3 + 5^{-3} ! This ignores the multiplication between 5−3 5^{-3} and 53 5^3 . Always apply the exponent rule am⋅an=am+n a^m \cdot a^n = a^{m+n} to multiply terms with the same base first.

Practice Quiz

Test your knowledge with interactive questions

Which of the following is equivalent to \( 100^0 \)?

FAQ

Everything you need to know about this question

Why does 5−3⋅53 5^{-3} \cdot 5^3 equal 1?

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When you multiply powers with the same base, you add the exponents: 5−3⋅53=5−3+3=50 5^{-3} \cdot 5^3 = 5^{-3+3} = 5^0 . Since any number to the power of 0 equals 1, we get 50=1 5^0 = 1 !

What does a negative exponent mean?

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A negative exponent means take the reciprocal: 5−3=153=1125 5^{-3} = \frac{1}{5^3} = \frac{1}{125} . It's the opposite of a positive exponent!

Do I need to calculate 53 5^3 to get the final answer?

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No! The answer is 53+1 5^3 + 1 in its simplified form. You don't need to calculate 53=125 5^3 = 125 unless specifically asked for a numerical value.

Why can't I just add the first 53 5^3 and the last 53 5^3 ?

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Because there's a multiplication between 5−3 5^{-3} and 53 5^3 ! You must handle multiplication before addition. The expression is 53+(5−3⋅53) 5^3 + (5^{-3} \cdot 5^3) , not 53+5−3+53 5^3 + 5^{-3} + 5^3 .

How do I remember the exponent multiplication rule?

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Think of it as collecting the same bases: x2⋅x3 x^2 \cdot x^3 means (x⋅x)⋅(x⋅x⋅x)=x2+3=x5 (x \cdot x) \cdot (x \cdot x \cdot x) = x^{2+3} = x^5 . You're just adding up how many times the base appears!

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