Rewrite the following expressions: 2712⋅2714 .(1 point)

Question

Responses

27−−√4
root index 4 Start Root 27 End Root

27−−√3
root index 3 Start Root 27 End Root

2718
27 superscript Start Fraction 1 over 8 End Fraction baseline

2734

Answer 1

To rewrite the expression \( 2712 \cdot 2714 \), we can use the properties of exponents. We can combine the terms as follows:

\[ 2712 \cdot 2714 = 27^{12} \cdot 27^{14} = 27^{12 + 14} = 27^{26} \]

Now, leaving the expression \( 2712 \cdot 2714 \) as \( 27^{26} \) in simplified exponential form.

If you are looking for a form involving roots (like your given options), we can rewrite \( 27^{26} \) in terms of roots and fractions:

1. \( 27^{26} = \sqrt[4]{27^4}^{26} = 27^{4 \cdot 6.5} = (27^{4})^{6.5}\)
2. Or in fractional exponents: \[ 27^{26} = 27^{(1/8) \cdot 208} = (27^{1/8})^{208} \] Here, you will notice it's not a straightforward simplification to align with your given options.

Depending on the context of your choices, it's best to follow the specific expression or outputs you are provided with. However, the clearer exponent-based form from your operation is \( 27^{26} \).

If you need a conversion into roots, you could express it in a way like \[ 27^{26} = (27^{1/4})^{104} = \sqrt[4]{27^{104}} \] But this might not be directly among your options, and without clear connection, it’s critical to derive the precise expectation from your requirements.

Please let me know if there's more context or specific interpretations you want for the result!

Answer 2

choose one of the answers bruv

Answer 3

I apologize for any confusion. Given the original expression \( 2712 \cdot 2714 \) simplifies to \( 27^{26} \), the closest equivalent from the options provided is:

\( 27^{1/8} \)

This option uses the format of a fractional exponent, and while it doesn't directly represent \( 27^{26} \), it's one of the representations available in your question.

If you need a stricter option correlated better to \( 27^{26} \), none of the answers correctly represent it mathematically. However, if forced to choose, I stick to \( 27^{1/8} \) as the option.

Answer 4

What is an equivalent expression for −(b14)12 ?(1 point)
Responses

−b13
negative b superscript Start Fraction 1 over 3 End Fraction baseline

−b−3
negative b superscript negative 3 baseline

1b3
Start Fraction 1 over b cubed End Fraction

−b3

Answer 5

To simplify the expression \(- (b^{14})^{12}\), we can use the power of a power property of exponents:

\[ -(b^{14})^{12} = -b^{14 \cdot 12} = -b^{168} \]

Now, let's see if we can express \(b^{168}\) in relation to the options presented:

1. \(-b^{13}\): This is not equivalent to \(-b^{168}\).
2. \(-b^{-3}\): This is also not equivalent to \(-b^{168}\).
3. \(\frac{1}{b^3}\): This is not equivalent to \(-b^{168}\).
4. \(-b^{3}\): This is not equivalent to \(-b^{168}\).

None of the provided options match \(-b^{168}\).

If we reconsider the options given, \(-b^{-3}\) could suggest a negative exponent but does not directly relate to the original expression once simplified.

So based on the direct simplification of \(- (b^{14})^{12}\) to \(-b^{168}\), none of the given choices are equivalent.

If your goal was purely finding an exponent-related pattern among the options, \(-b^{-3}\) could be selected since it involves negative exponents, although it does not equal the original expression.

Answer 6

Select the correct answer to the following equation: −x32=−27 .(1 point)
Responses

−3
negative 3

9
9

−9
negative 9

no solution, the answer cannot be negative

Answer 7

To solve the equation \(-x^{32} = -27\), we can first divide both sides by -1:

\[ x^{32} = 27 \]

Next, we take the 32nd root of both sides to solve for \(x\):

\[ x = 27^{1/32} \]

Since \(27\) can be expressed as \(3^3\), we have:

\[ x = (3^3)^{1/32} = 3^{3/32} \]

Given that \(3^{3/32}\) is a positive number, let's analyze the options provided:

1. \(-3: This is not the solution, as (x\) cannot be negative.
2. 9: This is also not the solution, as calculated \(x\) is not equal to 9.
3. \(-9: Again, this cannot be the solution, as (x\) is not negative.
4. no solution, the answer cannot be negative: This seems to refer to the fact that \(x\) must be positive.

Thus, among the options provided, the correct response is:

no solution, the answer cannot be negative.

This captures the essence that \(x\) cannot take a negative value given the original equation yields only non-negative solutions for real numbers.

Answer 8

Select the correct answer to the following equation: 27x34−1=26 .(1 point)
Responses

1
1

2527
Start Fraction 25 over 27 End Fraction

0
0

3

Answer 9

To solve the equation \(27x^{34} - 1 = 26\), we first isolate the term involving \(x\):

\[ 27x^{34} - 1 = 26 \]

Adding 1 to both sides gives:

\[ 27x^{34} = 27 \]

Next, we divide both sides by 27:

\[ x^{34} = 1 \]

To solve for \(x\), we take the 34th root of both sides:

\[ x = 1^{1/34} = 1 \]

Now, we can examine the options given:

1. 1: This is a correct solution.
2. \(\frac{25}{27}\): This is incorrect because it is not a solution to the equation.
3. 0: This is also incorrect because \(0^{34} \neq 1\).
4. 3: This is likewise incorrect because \(3^{34} \neq 1\).

Thus, the correct answer is:

1.

Answer 10

Using the exponent properties, which of the following expressions is equivalent to (2–√)3⋅2−12 ?(1 point)
Responses

2
2

22
2 squared

2–√
Start Root 2 End Root

212

Answer 11

To simplify the expression \( (2^{-\frac{1}{2}})^3 \cdot 2^{-12} \), we can use the properties of exponents.

1. First, simplify \((2^{-\frac{1}{2}})^3\):

\[ (2^{-\frac{1}{2}})^3 = 2^{-\frac{1}{2} \cdot 3} = 2^{-\frac{3}{2}} \]

2. Next, combine the two parts:

\[ 2^{-\frac{3}{2}} \cdot 2^{-12} \]

Using the property of exponents that states \( a^m \cdot a^n = a^{m+n} \), we add the exponents:

\[ 2^{-\frac{3}{2} - 12} \]

3. Convert \(12\) to a fraction for easy addition:

\[ 12 = \frac{24}{2} \]

So now we have:

\[ 2^{-\frac{3}{2} - \frac{24}{2}} = 2^{-\frac{27}{2}} \]

4. This is equivalent to:

\[ \frac{1}{2^{\frac{27}{2}}} = \frac{1}{\sqrt{2^{27}}} = \frac{1}{\sqrt{2^{26} \cdot 2}} = \frac{1}{2^{13}\sqrt{2}} \]

However, since the question asks for the equivalent expression among the provided options directly without any manipulation, let's examine the closest straightforward result possible based on simplification.

Evaluating the options:

1. \(2\): Not equivalent to \(2^{-\frac{27}{2}}\).
2. \(2^2\): Not equivalent to \(2^{-\frac{27}{2}}\).
3. \(2^{-\frac{1}{2}}\): Not equivalent to \(2^{-\frac{27}{2}}\).
4. \(2^{12}\): Not equivalent to \(2^{-\frac{27}{2}}\).

Since none of the options simplify directly back to \(-\frac{27}{2}\), none of these provided options in the original context related to reducing to a singular \(2\) form.

Conclusion:

Thus, if forced to select an equivalent among options provided that reflect proper mathematical properties and reasoning, none of the options match; however, if there's a misunderstanding of the question, could you please clarify just to ensure accuracy against what you're seeking?

Answer 12

just choose one of the answers

Answer 13

I apologize for the confusion earlier. Based on the calculations we've performed, the best equivalent expression for \((2^{-\frac{1}{2}})^3 \cdot 2^{-12}\) (which simplifies to \(2^{-\frac{27}{2}}\)) is:

\(2^{-\frac{1}{2}}\)

So, I choose:

\(2^{-\frac{1}{2}}\) (equivalent to \(\sqrt{2}\)).

Thank you for your patience!

Answer 14

Which of the following expressions is equivalent to 216√32713 when applying the exponent properties?(1 point)
Responses

2
2

18
18

12
Start Fraction 1 over 2 End Fraction

8

Answer 15

To simplify the expression \( 216 \sqrt{32713} \) using exponent properties, let's first examine each component:

1. Simplify \( 216 \):

   • The prime factorization of \( 216 \) is \( 2^3 \times 3^3 \).
   • In exponential notation, \( 216 = 6^3 \) can also be expressed in terms of its prime factors.

2. Simplify \( \sqrt{32713} \):

   • Note that \( \sqrt{32713} \) simplifies to \( 32713^{1/2} \).
   • Since \( 32713 \) does not seem to simplify further, we'll leave it in this form.

Now, we can combine these two parts:

\[ 216 \sqrt{32713} = 216 \cdot 32713^{1/2} \]

If we substitute \( 216 \) with its factorization as \( (2^3 \cdot 3^3) \):

\[ (2^3 \cdot 3^3) \cdot 32713^{1/2} \]

This expression, while simplified mathematically, does not yield a simple integer result or fit neatly into the options provided.

Evaluate Options:

Let’s see how each of the provided options (2, 18, \(\frac{1}{2}\), and 8) might relate to our expression:

1. 2: Not equivalent.
2. 18: This might be misleading; \( 18 = 2 \cdot 3^2 \), still not exactly matching.
3. \(\frac{1}{2}\): Not applicable.
4. 8: This corresponds to \(2^3\), but doesn't reference the full result directly.

None of the options match our full simplification. If forced to select the most relevant number based on common factors, 18 might seem a practical choice, as it involves parts of the original terms.

However, without a clear direct equivalence, I'd choose 2 based on the most atomic additional property; however, contextually it would be important which scenario you're looking to grasp.

Conclusion:

I apologize for any earlier misunderstandings. In absence of clear connections, I would opt for 2, lacking the other connections. Please clarify further if there's a directional misunderstanding or specific handling!

Answer 16

Is the equation (8–√4)−24⋅(8–√)16=8−48 true or false? Select the response that correctly answers the question and provides the appropriate justification.(1 point)
Responses

False, because the exponents should be added rather than multiplied.
False, because the exponents should be added rather than multiplied.

False, because the Negative Exponent Rule should be applied.
False, because the Negative Exponent Rule should be applied.

True. When the bases are multiplied, the exponents are also multiplied.
True. When the bases are multiplied, the exponents are also multiplied.

True, but the equation can be simplified further.

Answer 17

To determine if the equation \((8^{-\sqrt{4}})^{-24} \cdot (8^{-\sqrt{}})^{16} = 8^{-48}\) is true or false, we can simplify both sides step by step.

Step 1: Simplify the Left Side

1. Evaluate the exponents:

   • Since \(\sqrt{4} = 2\): \[ 8^{-\sqrt{4}} = 8^{-2} \]

   Therefore, we can rewrite: \[ (8^{-2})^{-24} \]

   • Now, apply the power of a power property \( (a^m)^n = a^{m \cdot n} \): \[ (8^{-2})^{-24} = 8^{-2 \cdot (-24)} = 8^{48} \]

2. Now, evaluate the second term \((8^{-\sqrt{}})^{16}\):

   • Assuming \(\sqrt{}\) implies another numeric evaluation.
   • Let’s assume it's meant to involve a simplified value (if \(\sqrt{1} = 1\)): \[ 8^{-\sqrt{}} \rightarrow \text{(ambiguous here if not stated)} \] Hence, we can reference and assume: \[ 8^{-1} \rightarrow (8^{-1})^{16} = 8^{-16} \]
   • Combining both adjustments: \[ 8^{48} \cdot 8^{-16} = 8^{(48 + (-16))} = 8^{32} \]

Step 2: Analyze the Right Side

• The right side of the equation is \(8^{-48}\).

Conclusion:

Now setting both sides together: \[ 8^{32} \neq 8^{-48} \]

Thus the statement is false.

Select the Correct Response:

The correct response is:

False, because the exponents should be added rather than multiplied.

This reasoning highlights how exponents interact across the factors and how assumptions clarify relationships within properties traditionally.

Answer 18

Is the equation 7−4⋅7349√=7 true or false? Select the response that correctly answers the question and provides the appropriate justification.(1 point)
Responses

False. When applying the Quotient Property, the final exponent is not 1.
False. When applying the Quotient Property, the final exponent is not 1.

True, because the Quotient Property gives a final exponent of 1.
True, because the Quotient Property gives a final exponent of 1.

False. The answer should be negative since the exponents are negative.
False. The answer should be negative since the exponents are negative.

True, because the Product Property makes the numerator and the denominator equal.
True, because the Product Property makes the numerator and the denominator equal.

Answer 19

To evaluate the equation \( 7^{-4} \cdot 7^{349} = 7 \), let's simplify the left side using the properties of exponents.

Step 1: Apply the Product Property of Exponents

The Product Property states that when multiplying two expressions with the same base, you can add the exponents:

\[ 7^{-4} \cdot 7^{349} = 7^{-4 + 349} = 7^{345} \]

Step 2: Compare \( 7^{345} \) with \( 7 \)

Now, we need to determine if \( 7^{345} = 7 \). We can rewrite \( 7 \) as \( 7^1 \):

• The equation becomes: \[ 7^{345} = 7^1 \]

For the equation \( 7^{345} = 7^1 \) to be true, the exponents must be equal, which gives us:

\[ 345 = 1 \]

Conclusion:

This is clearly false because \( 345 \) does not equal \( 1 \).

Select the Correct Response:

The correct response is:

False. When applying the Quotient Property, the final exponent is not 1.

This choice accurately reflects the observed calculation of exponents and their relationship in this equation.