Using the properties of exponents, which expression is equivalent to x16⋅x16 ?(1 point)

Question

Responses

x−−√
Start Root x End Root

x−−√3
root index 3 Start Root x End Root

1x√3
Start Fraction 1 over root index 3 Start Root x End Root End Fraction

x−−√36

please anser in word form

Answer 1

The expression \(x^{16} \cdot x^{16}\) can be simplified using the properties of exponents. Specifically, when multiplying two expressions with the same base, you add their exponents.

So, we have:

1. Combine the exponents: \( x^{16} \cdot x^{16} = x^{16 + 16} = x^{32} \).

Now, let's analyze the response options provided to find a match for \(x^{32}\):

• Root of x: This represents \(x^{1/2}\), which is not equivalent to \(x^{32}\).
• Root index 3 of x: This represents \(x^{1/3}\), which also is not equivalent to \(x^{32}\).
• Fraction 1 over root index 3 of x: This represents \(x^{-1/3}\), which is not equivalent to \(x^{32}\).
• Root of 36 multiplied by x: This option may represent a different operation and does not correspond to \(x^{32}\).

None of the given responses are equivalent to \(x^{32}\). Therefore, the final representation we are looking for, which is not listed among the options, is \(x^{32}\).

Thus, in word form, the answer is: "x raised to the power of thirty-two."

Answer 2

Using the properties of exponents, which expression is equivalent to xx34 ?(1 point)
Responses

x
x

x4
x superscript 4 baseline

x−−√4
root index 4 Start Root x End Root

1​x√4

Answer 3

To simplify the expression \(x^{x^{34}}\), we start by recognizing that the exponent is \(x^{34}\).

The expression \(x^{x^{34}}\) does not directly simplify to any of the choices as they are presented.

Let's evaluate the given options against the original expression \(x^{x^{34}}\):

1. \(x\): This represents \(x^1\), which is not equivalent to \(x^{x^{34}}\).

2. \(x^4\): This represents an expression where \(x\) is raised to the power of \(4\). It is not equivalent to \(x^{x^{34}}\).

3. \(\sqrt[4]{x}\): This represents \(x^{1/4}\), which is not equivalent to \(x^{x^{34}}\).

4. \(\frac{1}{\sqrt{4}}\): This is just a numerical value, specifically \( \frac{1}{2} \). This also does not relate to the original expression.

None of the options provided are equivalent to the expression \(x^{x^{34}}\).

Thus, none of the answers match the original expression \(x^{x^{34}}\). The original expression remains as is.

Answer 4

Solve the following equation with a rational exponent: 2x32=16 .(1 point)
Responses

4
4

no solution
no solution

84
84

8

Answer 5

To solve the equation \(2x^{32} = 16\), we can start by isolating \(x^{32}\).

Here are the steps:

1. Divide both sides by 2: \[ x^{32} = \frac{16}{2} \] \[ x^{32} = 8 \]

2. Now, rewrite 8 as a power of 2: \[ 8 = 2^3 \] So we have: \[ x^{32} = 2^3 \]

3. Now, take the 32nd root (which is the same as raising to the power of \(\frac{1}{32}\)) of both sides: \[ x = (2^3)^{1/32} \] Using the properties of exponents, we multiply the exponents: \[ x = 2^{3/32} \]

At this point, we evaluate which of the provided options matches \(2^{3/32}\):

• 4: This is \(2^2\).
• no solution: This implies there’s no possible \(x\).
• 84: This is not a power of 2.
• 8: This is \(2^3\).

The correct answer, \(2^{3/32}\), does not match any given responses exactly. Thus, the closest understanding is that there is indeed a valid solution \(x = 2^{3/32}\), but none of the provided options represent this solution.

If we check understanding based on the context of rational solutions, it seems there is no suitable match. Please consider the context of the answers carefully. If you need to select one from the options provided, one must note that the correct answer isn't represented.

Answer 6

Solve the following equation with a rational exponent: 5x43=80 .(1 point)
Responses

40.32
40.32

8
8

12
12

16

Answer 7

To solve the equation \(5x^{43} = 80\), we start by isolating \(x^{43}\).

Here are the steps:

1. Divide both sides by 5: \[ x^{43} = \frac{80}{5} \] \[ x^{43} = 16 \]

2. Now, we can express 16 as a power of 2: \[ 16 = 2^4 \] So we have: \[ x^{43} = 2^4 \]

3. Now, take the 43rd root of both sides (which is the same as raising to the power of \(\frac{1}{43}\)): \[ x = (2^4)^{\frac{1}{43}} \] Using the properties of exponents, we multiply the exponents: \[ x = 2^{\frac{4}{43}} \]

Now evaluating which of the given response options matches \(2^{\frac{4}{43}}\):

• 40.32: This is a decimal and not a power of 2.
• 8: This represents \(2^3\).
• 12: This does not represent a power of 2.
• 16: This represents \(2^4\), which is related but does not equal \(x\).

Since none of the given options correspond to the calculated \(x = 2^{\frac{4}{43}}\), it appears that there is no matching answer from the choices provided.

The solution \(x = 2^{\frac{4}{43}}\) is valid, but it does not match any of the provided options exactly. Therefore, please consider that the appropriate selection for the answer may not be available. If you had to choose from the options given, the closest in context might be 8, as it is a power of 2, but it does not reflect the correct value of \(x\).

Answer 8

Solve the following equation with a rational exponent: x23=64 .(1 point)
Responses

16
16

8
8

512
512

32

Answer 9

To solve the equation \(x^{23} = 64\), we can follow these steps:

1. First, recognize that \(64\) can be expressed as a power of \(2\): \[ 64 = 2^6 \] So, we can rewrite the equation as: \[ x^{23} = 2^6 \]

2. Next, we can solve for \(x\) by taking the 23rd root of both sides (which is equivalent to raising both sides to the power of \(\frac{1}{23}\)): \[ x = (2^6)^{\frac{1}{23}} \] Using the properties of exponents, we multiply the exponents: \[ x = 2^{\frac{6}{23}} \]

3. Now let's evaluate which of the response options matches \(x = 2^{\frac{6}{23}}\):

• 16: This is equal to \(2^4\).
• 8: This is equal to \(2^3\).
• 512: This is equal to \(2^9\).
• 32: This is equal to \(2^5\).

None of the provided response options correspond to \(x = 2^{\frac{6}{23}}\).

Since \(2^{\frac{6}{23}}\) is not one of the provided answers, the solution is valid but does not match any of the given choices. If you had to select one from the options, you should note that none reflect the correct value of \(x\).