Multiplication of Powers: Identify the greater 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 $(y^4)^3$.

To determine which expression has the greatest value, we apply the exponent rules to simplify each choice:

• For $x^3 \times x^4$, using the product rule: $x^3 \times x^4 = x^{3+4} = x^7$.
• For $(x^3)^5$, using the power of a power rule: $(x^3)^5 = x^{3 \times 5} = x^{15}$.
• $x^{10}$ is already in its simplest form.
• For $\frac{x^9}{x^2}$, using the quotient rule: $\frac{x^9}{x^2} = x^{9-2} = x^7$.

To identify the greater value, we compare the exponents:

• $x^7$ from choices 1 and 4.
• $x^{15}$ from choice 2.
• $x^{10}$ from choice 3.

The expression with the largest exponent is $(x^3)^5$ or $x^{15}$.

Therefore, the expression with the greatest value is $(x^3)^5$.

We start by simplifying each expression using the laws of exponents.

For the first expression $2^2 \cdot 2^{-3} \cdot 2^4$:

• Apply the multiplication of powers rule: $2^2 \cdot 2^{-3} = 2^{2 + (-3)} = 2^{-1}$.
• Now, multiply by $2^4$: $2^{-1} \cdot 2^4 = 2^{-1 + 4} = 2^3$.

Thus, the first expression simplifies to $2^3$.

For the second expression $2^3 \cdot 2^{-2} \cdot 2^5$:

• Apply the multiplication of powers rule: $2^3 \cdot 2^{-2} = 2^{3 + (-2)} = 2^1$.
• Now, multiply by $2^5$: $2^1 \cdot 2^5 = 2^{1 + 5} = 2^6$.

Thus, the second expression simplifies to $2^6$.

To compare $2^3$ and $2^6$, we recognize that $2^6$ is greater than $2^3$. Hence, the second expression is greater.

Thus, the correct answer is: $<$.

To solve this problem, we'll apply the multiplication of powers rule, which states that for the same base $a$, $a^m \times a^n = a^{m+n}$. Let's simplify each expression:

For the left side:

• $2^3 \times 2^4 \times 2^8$
• Add the exponents since the bases are the same: $3 + 4 + 8 = 15$
• The simplified form is $2^{15}$

For the right side:

• $2^2 \times 2^3 \times 2^{10}$
• Add the exponents as well: $2 + 3 + 10 = 15$
• The simplified form is $2^{15}$

Now, comparing the two sides: $2^{15}$ and $2^{15}$.

Since both are the same power of 2, we conclude that:

The correct sign to insert is $=$.

Therefore, the solution to the problem is $\boxed{=}$.

To solve this problem, we'll follow these steps:

• Step 1: Simplify the left-hand side expression.

• Step 2: Simplify the right-hand side expression.

• Step 3: Compare the results.

Let's work through each step:

Step 1: Simplifying the left-hand side expression:

The expression is $3^{-5} \times 3^{17} \times 3^{-7}$.

Use the rule of exponents for multiplication: $a^m \times a^n = a^{m+n}$.

Add the exponents: $-5 + 17 - 7 = 5$.

So, $3^{-5} \times 3^{17} \times 3^{-7} = 3^5$.

Step 2: Simplifying the right-hand side expression:

The expression is $3^2 \times 3^1 \times 3$.

This can be rewritten as $3^2 \times 3^1 \times 3^1$, since $3 = 3^1$.

Add the exponents: $2 + 1 + 1 = 4$.

So, $3^2 \times 3^1 \times 3 = 3^4$.

Step 3: Compare $3^5$ and $3^4$.

Since the base 3 is the same, compare the exponents: $5$ and $4$.

Since 5 > 4, it follows that 3^5 > 3^4.

Therefore, the inequality sign between the two expressions is >.

Thus, the correct answer to the initial problem is 3^{-5}\times3^{17}\times3^{-7} > 3^2\times3^1\times3.

To find which expression has the greatest value given $x > 1$, we will apply exponent rules:

• Expression (1): $x^2 \times x^9$
• Expression (2): $x^2 \times x^3$
• Expression (3): $x^{10} \times x^{-7}$
• Expression (4): $x \times x$

Now, let's simplify each expression:

• Expression (1): Using $x^a \times x^b = x^{a+b}$, we get $x^2 \times x^9 = x^{2+9} = x^{11}$.
• Expression (2): Similarly, $x^2 \times x^3 = x^{2+3} = x^5$.
• Expression (3): Thus, $x^{10} \times x^{-7} = x^{10-7} = x^3$.
• Expression (4): It becomes $x \times x = x^{1+1} = x^2$.

Now, compare the powers: $11, 5, 3,$ and $2$. Since $x > 1$, the greater the exponent, the greater the value of the expression. Thus, the expression with the largest power of $x$ is $x^{11}$ from expression (1).

Therefore, the expression with the largest value is $x^2 \times x^9$.

To solve this problem, we'll follow these steps:

• Simplify each expression using the rules of exponents.
• Compare the resulting powers since $c > 1$ implies that the expression with the highest power excites the largest value.

Now, let's work through each expression:
1. For choice (1) $c^2 \times c^{-3}$:
Applying the exponent rule $c^{2-3} = c^{-1}$.

2. For choice (2) $c^2 \times c^1$:
Using the exponent rule $c^{2+1} = c^3$.

3. For choice (3) $c^{-2} \times c^{-2}$:
Using the exponent rule $c^{-2-2} = c^{-4}$.

4. For choice (4) $(c)^4 \times c^1$:
Applying the exponent rule $c^{4+1} = c^5$.

Given that $c > 1$, the expression with the largest exponent will have the largest value. Comparing all the simplified expressions, we have exponents: $-1, 3, -4,$ and $5$.
Therefore, the expression with the greatest value is $(c)^4 \times c^1$, which corresponds to choice (4) since $c^5$ has the largest exponent.

To solve this problem, let's simplify and compare the given expressions one by one.

• Simplification of each expression:
• $b^3 \times b^5 \times b^{-2} = b^{3+5-2} = b^6$
• $b^7 \times b^4 = b^{7+4} = b^{11}$
• $(b)^3 \times b^4 = b^{3+4} = b^7$
• $b^{-3} \times b^6 = b^{-3+6} = b^3$

Next, we compare the simplified exponents:
- The first expression simplifies to $b^6$.
- The second expression simplifies to $b^{11}$.
- The third expression simplifies to $b^7$.
- The fourth expression simplifies to $b^3$.

Among these, $b^{11}$ is the greatest because exponent 11 is the highest. Since $b > 1$, greater exponents correspond to greater values.

Therefore, the expression with the greatest value is $b^7 \times b^4$, which corresponds to choice 2.

Let's systematically simplify both expressions and then compare them:

Simplifying the First Expression:

$\frac{7^2 \cdot 7^{-8}}{7^3 \cdot (-7)^4}$

• Apply Product of Powers Rule to the numerator: $7^2 \cdot 7^{-8} = 7^{2 + (-8)} = 7^{-6}$.

• Use Power of a Power Rule in the denominator for $(-7)^4 = (-1)^4 \cdot 7^4 = 7^4$ because $(-1)^4 = 1$.

• Simplify the denominator: $7^3 \cdot 7^4 = 7^{3+4} = 7^7$.

• Apply Quotient of Powers Rule: $\frac{7^{-6}}{7^7} = 7^{-6-7} = 7^{-13}$.

Simplifying the Second Expression:

$\frac{7^2 \cdot 7^{-9}}{7^3 \cdot (-7)^4}$

• Apply Product of Powers Rule to the numerator: $7^2 \cdot 7^{-9} = 7^{2 + (-9)} = 7^{-7}$.

• The denominator is the same as before: $7^3 \cdot 7^4 = 7^7$.

• Apply Quotient of Powers Rule: $\frac{7^{-7}}{7^7} = 7^{-7-7} = 7^{-14}$.

Comparison:

• The first expression simplifies to $7^{-13}$.

• The second expression simplifies to $7^{-14}$.

• Since -13 > -14,
  7^{-13} > 7^{-14}.

Therefore, the first expression is greater than the second expression. The correct choice is: > .