Value Comparison: Determining the Greater Magnitude

Question

Which value is greater?

Video Solution

Solution Steps

00:00 What is the biggest value?
00:04 According to the laws of exponents, multiplication with the same base(A) with exponents N,M
00:06 will be equal to the same base(A) with exponent (N+M)
00:10 Let's apply it to the question
00:14 We'll use the formula and combine the exponents
00:17 According to the laws of exponents, raising the base(A) with exponent(M) to the power of(N)
00:20 will give us the base(A) with an exponent that is the product of the powers(N,M)
00:23 Let's apply it to the question
00:27 We'll use the formula and multiply the exponents
00:34 According to the laws of exponents, when dividing with the same base(A)
00:37 with different exponents(M,N) where M is in the numerator
00:40 we get the same base(A) with exponent(M-N)
00:43 Let's apply it to the question and subtract the exponents
00:47 Always subtract the denominator's exponent from the numerator's exponent
00:52 Let's go through the answers and find the largest one
01:02 And this is the solution to the question

Step-by-Step Solution

To solve this problem, we need to simplify and compare the given expressions.

Let's simplify each:

  • y7×y2 y^7 \times y^2 :
    Using the product of powers rule, y7×y2=y7+2=y9 y^7 \times y^2 = y^{7+2} = y^9 .
  • (y4)3 (y^4)^3 :
    Using the power of a power rule, (y4)3=y4×3=y12 (y^4)^3 = y^{4 \times 3} = y^{12} .
  • y9 y^9 :
    This is already in its simplest form, y9 y^9 .
  • y11y4 \frac{y^{11}}{y^4} :
    Using the power of a quotient rule, y11y4=y114=y7 \frac{y^{11}}{y^4} = y^{11-4} = y^7 .

Now that all the expressions are in the form yn y^n , we can compare the exponents to see which is greatest: y9y^9, y12y^{12}, y9y^9, and y7y^7.

The expression with the highest power is y12 y^{12} , which corresponds to the choice (y4)3 (y^4)^3 .

Thus, the greater value among the choices is (y4)3 (y^4)^3 .

Answer

(y4)3 (y^4)^3