Power of a Power: Identify the greater value

To solve the problem, we need to simplify and compare the following two expressions:

$\left(\left(x \times y\right)^5\right)^{12} \quad \text{and} \quad \left(\left(\left(x \times y\right)^{10}\right)^2\right)^3$

Let's simplify each expression:

• The first expression $\left(\left(x \times y\right)^5\right)^{12}$.
  Using the power of a power rule, $\left(a^m\right)^n = a^{m \cdot n}$, we have:
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$\left(\left(x \times y\right)^5\right)^{12} = \left(x \times y\right)^{5 \cdot 12} = \left(x \times y\right)^{60}$

• The second expression $\left(\left(\left(x \times y\right)^{10}\right)^2\right)^3$.
  Similarly, apply the power of a power rule twice:

$\left(\left(\left(x \times y\right)^{10}\right)^2\right)^3 = \left(\left(x \times y\right)^{10 \cdot 2}\right)^3$

$= \left(x \times y\right)^{20 \cdot 3} = \left(x \times y\right)^{60}$

After simplification, both expressions become $\left(x \times y\right)^{60}$.

Therefore, the relationship between the two expressions is:

$\left(x \times y\right)^{60} = \left(x \times y\right)^{60}$

=

Thus, the correct choice from the provided options is:

: =

I am confident that the solution is correct, as both expressions simplify to the same value.

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

Let's simplify each:

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

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

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

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

To determine which value is greater, let's simplify each choice:

Choice 1: $(a^2)^4$
By using the power of a power rule: $(x^m)^n = x^{m \times n}$, it simplifies to:
$(a^2)^4 = a^{2 \times 4} = a^8$.

Choice 2: $a^2 + a^0$
Evaluate using the zero exponent rule, $a^0 = 1$:
This expression becomes $a^2 + 1$.

Choice 3: $a^2 \times a^1$
Apply the product of powers rule: $x^m \times x^n = x^{m+n}$:
This simplifies to $a^{2+1} = a^3$.

Choice 4: $\frac{a^{14}}{a^9}$
Apply the quotient of powers rule: $\frac{x^m}{x^n} = x^{m-n}$:
This simplifies to $a^{14-9} = a^5$.

Now, let's compare these simplified forms:
We have $a^8$, $a^2 + 1$, $a^3$, and $a^5$.

For $a > 1$, exponential functions grow rapidly, thus:
- $a^8$ is greater than $a^5$.
- $a^8$ is greater than $a^3$.
- $a^8$ is greater than $a^2 + 1$ for sufficiently large $a$.

Thus, the expression with the highest power, and therefore the greatest value, is $(a^2)^4$.

To determine which of the given expressions is the greatest, we will use the relevant exponent rules to simplify each one:

• Simplify $y^7 \times y^2$:
  Using the Product of Powers rule, we have $y^7 \times y^2 = y^{7+2} = y^9$.
• Simplify $(y^4)^3$:
  Using the Power of a Power rule, we have $(y^4)^3 = y^{4 \times 3} = y^{12}$.
• Simplify $y^9$:
  This expression is already simplified and is $y^9$.
• Simplify $\frac{y^{11}}{y^4}$:
  Using the Division of Powers rule, we have $\frac{y^{11}}{y^4} = y^{11-4} = y^7$.

After simplifying, we compare the powers of $y$ from each expression:

• $y^9$ from $y^7 \times y^2$
• $y^{12}$ from $(y^4)^3$
• $y^9$ from $y^9$
• $y^7$ from $\frac{y^{11}}{y^4}$

Clearly, $y^{12}$ is the largest power among the expressions, meaning that $(y^4)^3$ is the greatest value.

Therefore, the correct choice is $(y^4)^3$.