Select all the expressions that are equivalent to 3^6 * 9^6. Multi-select Choices: (A)1/27^-6 (B)27^12 (C)1/27^6 (D)27^36

Question

Select all the expressions that are equivalent to 3^6 * 9^6.   Multi-select Choices: (A)1/27^-6 (B)27^12 (C)1/27^6 (D)27^36

Bytelearn · Accepted Answer

Recognize 9 as Power of 3: We need to simplify the expression 3^6 * 9^6 to find equivalent expressions. First, we recognize that 9 is a power of 3, specifically 9 = 3^2. So, we can rewrite 9^6 as (3^2)^6.Apply Power of a Power Rule: Now we apply the power of a power rule, which states that (a^b)^c = a^(b*c). So, (3^2)^6 becomes 3^(2*6) or 3^12.Multiply Exponents of Same Base: Next, we multiply the exponents of the same base in the original expression. So, 3^6 * 3^12 becomes 3^(6+12) or 3^18.Compare to Choices: Now we have simplified the original expression to 3^18. We will compare this to each of the choices to see which are equivalent.Simplify Choice (A): For choice (A) 1/27^-6, we recognize that 27 is 3^3. So, 1/27^-6 is the same as 1/(3^3)^-6. Applying the negative exponent rule, which states that 1/a^-b = a^b, we get (3^3)^6.Simplify Choice (B): Applying the power of a power rule again, (3^3)^6 becomes 3^(3*6) or 3^18. So, choice (A) is equivalent to 3^18.Simplify Choice (C): For choice (B) 27^12, we again recognize that 27 is 3^3. So, 27^12 is the same as (3^3)^12. Applying the power of a power rule, (3^3)^12 becomes 3^(3*12) or 3^36. This is not equivalent to 3^18.Simplify Choice (D): For choice (C) 1/27^6, we have 1 divided by 27 raised to the 6th power. As before, 27 is 3^3, so 1/27^6 is the same as 1/(3^3)^6. This simplifies to 1/3^18, which is the reciprocal of 3^18, not equivalent to 3^18.For choice (D) 27^36, we have 27 raised to the 36th power. Since 27 is 3^3, 27^36 is the same as (3^3)^36. Applying the power of a power rule, (3^3)^36 becomes 3^(3*36) or 3^108. This is not equivalent to 3^18.

Select all the expressions that are equivalent to $3^6 \times 9^6$ . $\newline$ Multi-select Choices: $\newline$ (A) $\frac{1}{27^{-6}}$ $\newline$ (B) $27^{12}$ $\newline$ (C) $\frac{1}{27^6}$ $\newline$ (D) $27^{36}$

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Select all the expressions that are equivalent to 36×963^6 \times 9^636×96.\newlineMulti-select Choices:\newline(A) 127−6\frac{1}{27^{-6}}27−61​\newline(B) 271227^{12}2712\newline(C) 1276\frac{1}{27^6}2761​\newline(D) 273627^{36}2736

Select all the expressions that are equivalent to $3^6 \times 9^6$ . $\newline$ Multi-select Choices: $\newline$ (A) $\frac{1}{27^{-6}}$ $\newline$ (B) $27^{12}$ $\newline$ (C) $\frac{1}{27^6}$ $\newline$ (D) $27^{36}$