What is the correct simplification of 243 3/5

Question

?(1 point)
Responses

81
81

9
9

27
27

3

Answer 1

To simplify \( 243 \frac{3}{5} \), we first convert the mixed number into an improper fraction or a decimal.

First, convert \( 243 \frac{3}{5} \) into an improper fraction:

\[ 243 \frac{3}{5} = \frac{(243 \times 5) + 3}{5} = \frac{1215 + 3}{5} = \frac{1218}{5} \]

Next, we want to see if we can simplify \( 243 \frac{3}{5} \) or find a number that this simplifies to, or a value that it represents roughly.

However, examining the options you provided:

• 81
• 9
• 27
• 3

We can see that \( 243 = 3^5 \) and breaking it down:

• \( 243 = 3 \times 81 \)
• \( 81 = 3^4 \)
• \( 27 = 3^3 \)

And since none of the options represent \( 243 \frac{3}{5} \) directly, let's check which value is associated with 243 when considering the whole integer part and ignoring the fractional part:

The integer part is 243 and from the responses, only 81, 9, 27, and 3 are given.

Realizing \( 243 \) can be simplified down to \( 27 \) in terms of powers of \( 3 \):

Since, \( \frac{243}{9} = 27 \).

Thus, the closest association considering possible simplifications would lead us to choose:

27

Answer 2

How can you rewrite 25 1/4
using a root?(1 point)
Responses

1/4−−√25
root index 25 Start Root Start Fraction 1 over 4 End Fraction End Root

25−−√1/4
root index Start Fraction 1 over 4 End Fraction Start Root 25 End Root

254−−−√1
root index 1 Start Root 25 superscript 4 baseline End Root

25−−√4

Answer 3

To rewrite \( 25 \frac{1}{4} \) using a square root, we can express the mixed number as an improper fraction and then apply the properties of square roots.

First, convert \( 25 \frac{1}{4} \) to an improper fraction:

\[ 25 \frac{1}{4} = \frac{(25 \times 4) + 1}{4} = \frac{100 + 1}{4} = \frac{101}{4} \]

Now, we can find different ways to express \( \frac{101}{4} \) using square roots.

Noting that:

\[ \frac{101}{4} = \frac{101}{\sqrt{4}} = \frac{\sqrt{101}}{\sqrt{4}} = \frac{\sqrt{101}}{2} \]

However, we are looking for a direct option among your provided responses.

Let's evaluate the given options:

1. \( \sqrt{25} \cdot \frac{1}{4} \)
2. \( \sqrt{25} \cdot \sqrt{\frac{1}{4}} \)
3. \( 25^{\frac{1}{4}} \cdot \sqrt{1} \)
4. \( \sqrt{25} \cdot 4 \)

The best interpretation is taking the square root of 25 multiplied by the square root of \( \frac{1}{4} \):

So, the correct response is \( \sqrt{25} \cdot \sqrt{\frac{1}{4}} \).

Answer 4

a,b,c or d

Answer 5

Given the options you provided, here is the correct interpretation of \( 25 \frac{1}{4} \) using square roots:

1. \( \sqrt{25} \cdot \frac{1}{4} \)
2. \( \sqrt{25} \cdot \sqrt{\frac{1}{4}} \)
3. \( 25^{\frac{1}{4}} \cdot \sqrt{1} \)
4. \( \sqrt{25} \cdot 4 \)

The best candidate that represents \( 25 \frac{1}{4} \) directly using roots would be:

B: \( \sqrt{25} \cdot \sqrt{\frac{1}{4}} \)

This can be simplified to:

\[ \sqrt{25 \cdot \frac{1}{4}} = \sqrt{\frac{25}{4}} = \frac{5}{2} \]

Which still relates back to \( 25 \frac{1}{4} \) conceptually.

So, the answer is B.

Answer 6

What is the simplest form of 25 1/2
?(1 point)
Responses

12.5
12.5

25−−√1/2
root index Start Fraction 1 over 2 End Fraction Start Root 25 End Root

25−−√
Start Root 25 End Root

5

Answer 7

To find the simplest form of \( 25 \frac{1}{2} \), let's first convert the mixed number to an improper fraction or decimal.

1. Convert \( 25 \frac{1}{2} \):

\[ 25 \frac{1}{2} = 25 + \frac{1}{2} = 25.5 \text{ or } \frac{51}{2} \]

2. Looking at the options provided:

• A: 12.5
• B: \( 25 \sqrt{\frac{1}{2}} \)
• C: \( \sqrt{25} \)
• D: 5

Since \( 25 \frac{1}{2} = 25.5 \) or \( \frac{51}{2} \), let's examine the options for correctness:

A: 12.5 does not match.

B: \( 25 \sqrt{\frac{1}{2}} \) simplifies to \( 25 \cdot \frac{1}{\sqrt{2}} \), which is not \( 25.5 \).

C: \( \sqrt{25} = 5 \); again, it does not match \( 25.5 \).

D: 5 does not match either.

None of the provided answers express \( 25 \frac{1}{2} \) completely or in a simplified form. However, since we are looking for the simplest numerical value from what you have given, it's most accurate to note that

if you were to simplify \( \sqrt{25} \), it would yield D: 5, which is the actual square root of the number 25.

But for the expression of \( 25 \frac{1}{2} \) specifically, none of the answers accurately simplify to 25.5 or \( \frac{51}{2} \).

Yet, based on the question asking for the simplest form of numerically related options available, you would most closely align with C: \( \sqrt{25} \) since it gives a correct answer of \( 5 \).

Final answer: C: \( \sqrt{25} \).

Answer 8

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

2734
27 superscript Start Fraction 3 over 4 End Fraction baseline

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

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

2718

Answer 9

Rewrite the following expressions: 27 1/2⋅27 1/4
.(1 point)
Responses

27 3/4
27 superscript Start Fraction 3 over 4 End Fraction baseline

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

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

27 1/8

Answer 10

To simplify the expression \( 27^{1/2} \cdot 27^{1/4} \), we can use the properties of exponents. When multiplying numbers with the same base, we add the exponents:

\[ 27^{1/2} \cdot 27^{1/4} = 27^{\frac{1}{2} + \frac{1}{4}} \]

To add \(\frac{1}{2}\) and \(\frac{1}{4}\), we can express \(\frac{1}{2}\) as \(\frac{2}{4}\):

\[ \frac{1}{2} + \frac{1}{4} = \frac{2}{4} + \frac{1}{4} = \frac{3}{4} \]

Thus, we can rewrite the expression as:

\[ 27^{3/4} \]

Now, examining the provided options:

1. \( 27^{3/4} \)
2. \( 27^{\frac{3}{4}} \)
3. \( 27^{\sqrt{4}} \)
4. \( 27^{\sqrt{3}} \)
5. \( 27^{1/8} \)

The correct rewrite of the expression \( 27^{1/2} \cdot 27^{1/4} \) is:

A: \( 27^{3/4} \) (or B depending on how fractions are represented in your options).

Both A and B mean the same (the formatting may differ).

So, the answer is 27^{3/4} (or you could mark B as well).

Answer 11

What is an equivalent expression for −(b1/4)^12
?(1 point)
Responses

−b^−3
negative b superscript negative 3 baseline

1/b3
Start Fraction 1 over b cubed End Fraction

−b^1/3
negative b superscript Start Fraction 1 over 3 End Fraction baseline

−b3

Answer 12

To simplify the expression \(- (b^{1/4})^{12}\), we can use the power of a power property of exponents. The property states that:

\[ (a^m)^n = a^{m \cdot n} \]

Applying this to our expression:

\[

• (b^{1/4})^{12} = - b^{(1/4) \cdot 12} = - b^{3} \]

Now let's evaluate the provided options:

1. −b^−3: This implies \(-\frac{1}{b^3}\), which is not correct.
2. 1/b^3: This implies \(\frac{1}{b^3}\), which is not correct.
3. −b^{1/3}: This is incorrect since we simplified to \( -b^{3} \).
4. −b^3: This is correct since it matches our simplified expression.

Thus, the equivalent expression for \(- (b^{1/4})^{12}\) is:

−b^3.