Which expression is equivalent to 3^6 * 9^6? Choices: (A)27^6 (B)1/27^6 (C)27^12 (D)27^36

Identify Bases and Exponents: Understand the given expression and identify the bases and exponents. We have the expression 3^6 * 9^6. Here, 3 and 9 are the bases, and both are raised to the power of 6.Recognize Power of 3: Recognize that 9 is a power of 3. Since 9 is 3^2, we can rewrite 9^6 as (3^2)^6.Apply Power of a Power Rule: Apply the power of a power rule. Using the power of a power rule, which states that (a^b)^c = a^(b*c), we can simplify (3^2)^6 to 3^(2*6) or 3^12.Combine Same Base Expressions: Combine the expressions with the same base. Now we have 3^6 * 3^12. Since the bases are the same, we can add the exponents: 3^(6+12) = 3^18.Recognize 3^18 Relationship: Recognize that 3^18 can be written as (3^6)^3. We can express 3^18 as (3^6)^3 because 6*3 equals 18.Simplify Further: Simplify the expression further. Since we know that 3^6 is 3^6, we can rewrite (3^6)^3 as 27^3, because 3^3 equals 27.

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Math Problems

Grade 8

Identify equivalent expressions involving exponents I

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Q. Which expression is equivalent to

3^6 \times 9^6

\newline

Choices:

\newline

(A)

27^6

\newline

(B)

\frac{1}{27^6}

\newline

(C)

27^{12}

\newline

(D)

27^{36}

Identify Bases and Exponents: Understand the given expression and identify the bases and exponents. $\newline$ We have the expression $3^6 \times 9^6$ . Here, $3$ and $9$ are the bases, and both are raised to the power of $6$ .
Recognize Power of $3$ : Recognize that $9$ is a power of $3$ . Since $9$ is $3^2$ , we can rewrite $9^6$ as $(3^2)^6$ .
Apply Power of a Power Rule: Apply the power of a power rule. $\newline$ Using the power of a power rule, which states that $(a^b)^c = a^{b*c}$ , we can simplify $(3^2)^6$ to $3^{2*6}$ or $3^{12}$ .
Combine Same Base Expressions: Combine the expressions with the same base. $\newline$ Now we have $3^6 \times 3^{12}$ . Since the bases are the same, we can add the exponents: $3^{6+12} = 3^{18}$ .
Recognize $3^{18}$ Relationship: Recognize that $3^{18}$ can be written as $(3^6)^3$ . We can express $3^{18}$ as $(3^6)^3$ because $6*3$ equals $18$ .
Simplify Further: Simplify the expression further. $\newline$ Since we know that $3^6$ is $3^6$ , we can rewrite $(3^6)^3$ as $27^3$ , because $3^3$ equals $27$ .