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 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 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 $\boxed{=}$.

We want to calculate each of the expressions separately, however - in order to do this more efficiently, we will first deal with the multiplication terms between the roots in both expressions separately:

a. Let's start with the left expression, the multiplication of roots in this expression is:

$\sqrt{10}\cdot\sqrt{10}$

We'll apply the laws of exponents in order to simplify this expression, noting that the expression is actually multiplying the number by itself and therefore can be written as a term to the second power:

$\sqrt{10}\cdot\sqrt{10}=(\sqrt{10})^2$

Now let's recall the definition of root as a power:

$\sqrt[n]{a}=a^{\frac{1}{n}}$

And the law of exponents for power to power:

$(a^m)^n=a^{m\cdot n}$

Let's apply these two laws and calculate the value of the above expression:

$(\sqrt{10})^2 =(10^{\frac{1}{2}})^2=10^{\frac{1}{2}\cdot2}=10^1=10$

Where in the first step we converted the root in parentheses to a half power using the definition of root as a power mentioned earlier, and in the next step we applied the law of power to power that was also mentioned earlier, then we simplified the expression.

b. Let's continue to the multiplication of roots in the right expression:

$\sqrt{5}\cdot\sqrt{20}$

In addition to the definition of root as a power mentioned earlier, let's also recall the law of exponents for powers in parentheses where terms are multiplied but in the opposite direction:

$x^n\cdot y^n=(x\cdot y)^n$

The literal interpretation of this law in the direction given here is that a multiplication between two terms with equal power exponents can be written as a multiplication between the bases in parentheses raised to that same power,

Let's return to the expression in question and apply both laws of exponents mentioned:

$\sqrt{5}\cdot\sqrt{20} =5^{\frac{1}{2}}\cdot20^{\frac{1}{2}}=(5\cdot20)^{\frac{1}{2}}$

Where in the first step we converted the roots to half powers using the definition of root as a power, and in the next step we applied the last mentioned law of exponents in its specified direction, since both terms in the multiplication here have the same power,

Let's continue and simplify the expression we got:

$(5\cdot20)^{\frac{1}{2}} =100^{\frac{1}{2}}=\sqrt{100}=10$

Where in the first step we calculated the value of the multiplication in parentheses, in the next step we returned to writing roots using the definition of root as power, but in the opposite direction, in the final step we calculated the numerical value of the root,

Let's summarize a and b above, we got that:

$\sqrt{10}\cdot\sqrt{10}=(\sqrt{10})^2 =10$ and

$\sqrt{5}\cdot\sqrt{20} =(5\cdot20)^{\frac{1}{2}} =\sqrt{100}=10$,

Let's return to the original problem and use this information:

$3^2+\sqrt{10}\cdot\sqrt{10}\text{ }_{\textcolor{red}{—}\text{ }}2^3+\sqrt{5}\cdot\sqrt{20}:5 \\ \downarrow\\ 3^2+10\text{ }_{\textcolor{red}{—}\text{ }}2^3+10:5$

Let's continue and handle both expressions together, in the left expression we'll first calculate the value of the term in the power and then the result of the addition,

And in the right expression we'll first calculate the result of the term in the power and the result of the division operation and add between the results:

$3^2+10\text{ }_{\textcolor{red}{—}\text{ }}2^3+10:5 \\ \downarrow\\ 9+10\text{ }_{\textcolor{red}{—}\text{ }}8+2 \\ \downarrow\\ 19\text{ }_{\textcolor{red}{—}\text{ }}10$

Therefore the left expression gives a higher result, meaning the trend between the expressions is:

19>10

Therefore the correct answer is answer B.

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 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 $(x^3)^5$.

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: > .

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, 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.

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.