Simplify the Expression: a^-3 × y^7 × a^-4 × y^-5 Using Exponent Rules

Exponent Rules with Negative Powers

Reduce the following equation:

a−3×y7×a−4×y−5= a^{-3}\times y^7\times a^{-4}\times y^{-5}=

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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 laws of exponents, the multiplication of exponents with equal bases (A)
00:06 equals the same base raised to the sum of the exponents (N+M)
00:11 Let's identify the equal bases
00:15 Let's apply this formula to our exercise, one operation at a time
00:20 We'll maintain the base and add together the exponents
00:32 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:

a−3×y7×a−4×y−5= a^{-3}\times y^7\times a^{-4}\times y^{-5}=

2

Step-by-step solution

To simplify the given expression a−3×y7×a−4×y−5 a^{-3} \times y^7 \times a^{-4} \times y^{-5} , we will apply the exponent rules.

First, let's handle the terms involving the base a a :

a−3×a−4 a^{-3} \times a^{-4}

According to the rule xm×xn=xm+n x^m \times x^n = x^{m+n} , we add the exponents:

a−3×a−4=a−3+(−4)=a−7 a^{-3} \times a^{-4} = a^{-3 + (-4)} = a^{-7}

Next, consider the terms involving the base y y :

y7×y−5 y^7 \times y^{-5}

Using the same exponent rule:

y7×y−5=y7+(−5)=y2 y^7 \times y^{-5} = y^{7 + (-5)} = y^2

The entire expression now becomes:

a−7×y2 a^{-7} \times y^2

Thus, the simplified form of the given expression is a−7×y2 a^{-7} \times y^2 .

3

Final Answer

a−7×y2 a^{-7}\times y^2

Key Points to Remember

Essential concepts to master this topic
  • Product Rule: When multiplying same bases, add the exponents together
  • Technique: Group like bases: a−3×a−4=a−3+(−4)=a−7 a^{-3} \times a^{-4} = a^{-3+(-4)} = a^{-7}
  • Check: Verify each base separately: a−7×y2 a^{-7} \times y^2 has correct exponents ✓

Common Mistakes

Avoid these frequent errors
  • Multiplying exponents instead of adding them
    Don't multiply the exponents like a−3×a−4=a12 a^{-3} \times a^{-4} = a^{12} = wrong answer! This confuses the product rule with the power rule. Always add exponents when multiplying same bases: a−3×a−4=a−7 a^{-3} \times a^{-4} = a^{-7} .

Practice Quiz

Test your knowledge with interactive questions

\( (3\times4\times5)^4= \)

FAQ

Everything you need to know about this question

Why do I add the exponents when I'm multiplying?

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The product rule says xm×xn=xm+n x^m \times x^n = x^{m+n} . Think of it this way: a2×a3=(a⋅a)×(a⋅a⋅a)=a5 a^2 \times a^3 = (a \cdot a) \times (a \cdot a \cdot a) = a^5 . You're combining all the factors!

What happens when I add negative exponents?

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Just follow normal addition rules! −3+(−4)=−7 -3 + (-4) = -7 , so a−3×a−4=a−7 a^{-3} \times a^{-4} = a^{-7} . Negative plus negative equals more negative.

Do I need to rearrange the terms first?

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It helps! Group like bases together: a−3×y7×a−4×y−5 a^{-3} \times y^7 \times a^{-4} \times y^{-5} becomes (a−3×a−4)×(y7×y−5) (a^{-3} \times a^{-4}) \times (y^7 \times y^{-5}) for easier calculation.

Can I simplify a−7 a^{-7} further?

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Yes! a−7=1a7 a^{-7} = \frac{1}{a^7} if you prefer positive exponents. But a−7×y2 a^{-7} \times y^2 is already in simplified form as requested.

What if I get confused about which exponents to add?

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Only add exponents for the same base! In this problem: add the a a exponents together (−3+(−4) -3 + (-4) ) and the y y exponents together (7+(−5) 7 + (-5) ).

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