Simplify the Expression: 5^y × 3^y Using Exponent Properties

Exponent Properties with Same Power Multiplication

Insert the corresponding expression:

5y×3y= 5^y\times3^y=

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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, a product raised to a power (N)
00:07 Equals the product where each factor is raised to the same power (N)
00:11 We will apply this formula to our exercise
00:19 This is the solution

Step-by-step written solution

Follow each step carefully to understand the complete solution
1

Understand the problem

Insert the corresponding expression:

5y×3y= 5^y\times3^y=

2

Step-by-step solution

To solve this problem, we'll simplify the expression by applying the properties of exponents:

  • Step 1: Recognize that both bases, 55 and 33, have the same exponent yy.
  • Step 2: Apply the power of a product rule in reverse: for terms ay×bya^y \times b^y, this simplifies to (a×b)y(a \times b)^y.
  • Step 3: Replace aa with 55 and bb with 33 to get (5×3)y(5 \times 3)^y.

Let's work through the solution with these steps:

Given the expression 5y×3y5^y \times 3^y, both terms share the same exponent yy. Therefore, we can combine them by multiplying the bases and keeping the common exponent:

(5×3)y (5 \times 3)^y

This simplification follows directly from the rule of exponents, which states an×bn=(a×b)na^n \times b^n = (a \times b)^n when nn is the same for both terms.

Therefore, the simplified expression is (5×3)y \left(5 \times 3\right)^y .

3

Final Answer

(5×3)y \left(5\times3\right)^y

Key Points to Remember

Essential concepts to master this topic
  • Rule: When multiplying terms with same exponents: an×bn=(a×b)n a^n \times b^n = (a \times b)^n
  • Technique: Multiply the bases and keep the common exponent: 5y×3y=(5×3)y 5^y \times 3^y = (5 \times 3)^y
  • Check: Test with specific values: if y=2 y = 2 , then 52×32=25×9=225=152 5^2 \times 3^2 = 25 \times 9 = 225 = 15^2

Common Mistakes

Avoid these frequent errors
  • Adding exponents instead of multiplying bases
    Don't write 5y×3y=52y 5^y \times 3^y = 5^{2y} or 32y 3^{2y} ! This confuses the product rule with the power rule and gives completely wrong answers. Always multiply the bases when the exponents are the same: an×bn=(a×b)n a^n \times b^n = (a \times b)^n .

Practice Quiz

Test your knowledge with interactive questions

\( (4^2)^3+(g^3)^4= \)

FAQ

Everything you need to know about this question

Why can I combine the bases when the exponents are the same?

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This works because of the distributive property of exponents! When you have 5y×3y 5^y \times 3^y , you're really multiplying y copies of 5 times y copies of 3, which equals y copies of (5×3).

What if the exponents were different, like 52×33 5^2 \times 3^3 ?

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You cannot combine the bases when exponents are different! 52×33 5^2 \times 3^3 stays as 25×27=675 25 \times 27 = 675 . The rule only works when exponents match exactly.

Does this work with more than two terms?

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Yes! For example: 2x×3x×4x=(2×3×4)x=24x 2^x \times 3^x \times 4^x = (2 \times 3 \times 4)^x = 24^x . As long as all exponents are identical, you can multiply all the bases together.

How is this different from 52×53 5^2 \times 5^3 ?

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That's a completely different rule! When the bases are the same but exponents differ, you add the exponents: 52×53=52+3=55 5^2 \times 5^3 = 5^{2+3} = 5^5 . Here we have different bases with same exponents.

Can I simplify (5×3)y (5 \times 3)^y further?

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You can write it as 15y 15^y if you want, but (5×3)y (5 \times 3)^y is already considered simplified in terms of exponent properties. Both forms are correct!

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