? Enter your response in fraction form.(1 point)
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11 answers
The expression \( 5 - \sqrt{3} \) can be rewritten in terms of rational exponents. The square root can be expressed using rational exponents as follows:
\[ \sqrt{3} = 3^{1/2} \]
Thus, the expression \( 5 - \sqrt{3} \) is equivalent to:
\[ 5 - 3^{1/2} \]
However, if you’re asking specifically about the "rational exponent" related to the square root part, the rational exponent for \( \sqrt{3} \) is \( \frac{1}{2} \).
Therefore, the rational exponent in fraction form is:
\[ \frac{1}{2} \]
The expression \(5 - \sqrt{3}\) does not directly correspond to \(5^?\) as it is not expressed in terms of a power of 5. However, if you need to express the square root of 3 in terms of a power, you can recognize that:
\[ \sqrt{3} = 3^{1/2} \]
And thus:
\[ 5 - \sqrt{3} = 5 - 3^{1/2} \]
If you're looking for a way to express the entire expression \(5 - \sqrt{3}\) in terms of a power of 5 specifically, there isn't a simple rational exponent for \(5^{?}\) that equals \(5 - \sqrt{3}\) since it represents a linear combination of a constant and a square root.
However, if you were looking for the exponent for \(5\) in terms of the original question about the rational exponent, please clarify what you mean by \(5^?\). If you meant to find a specific transformation related to \(5\), please provide more context.
To express \( 5 - \sqrt{3} \) in terms of rational exponents, let's focus on the term \( \sqrt{3} \).
The expression \( \sqrt{3} \) can be written using a rational exponent:
\[ \sqrt{3} = 3^{1/2} \]
To represent \(5 - \sqrt{3}\) in a way that highlights the exponent used, we can clarify that \(5\) can be expressed as \(5^1\). Hence, when you refer to \(5^?\) in relation to \(5 - \sqrt{3}\), this specific expression does not lead directly to a single exponent that combines both parts because \(5 - \sqrt{3}\) is not represented as a single exponent of 5.
If we isolate the term \( \sqrt{3} \):
The equivalent expression you might be looking for simplifies solely the square root:
\[ 5^{1} - 3^{1/2} \]
If you need the "rational exponent" for \(5^?\), the exponent of \(5\) in the form \(5^{?}\) is simply \(? = 1\).
For the square root term \(\sqrt{3}\):
\[ 3^{1/2} \]
Here, if you are looking for a rational exponent representation only for \(\sqrt{3}\), it is \(\frac{1}{2}\).
But given your question, if we are looking for the representation of just \(5\), then:
\[ \text{For } 5: \quad ? = 1 \]
Please clarify if you're looking for something different, as the original request might suggest a need for further elaboration.
To simplify the expression \( \frac{8\sqrt{3}}{2\sqrt{2}} \) and express it in terms of \( 2^? \), we can follow these steps:
The coefficient \( \frac{8}{2} = 4 \).
We can write \( 4 \) as \( 2^2 \).
Now we address the square roots:
\[ \sqrt{3} \text{ remains as } \sqrt{3} \] \[ \sqrt{2} \text{ remains as } \sqrt{2} \]
So our expression now looks like:
\[ \frac{2^2\sqrt{3}}{\sqrt{2}} \]
We can express \(\sqrt{2}\) as \(2^{1/2}\):
\[ = 2^2 \cdot \frac{\sqrt{3}}{2^{1/2}} \] \[ = 2^2 \cdot \sqrt{3} \cdot 2^{-1/2} \]
Using the property of exponents where \(a^m \cdot a^n = a^{m+n}\):
\[ = 2^{2 - \frac{1}{2}} \cdot \sqrt{3} \] \[ = 2^{\frac{4}{2} - \frac{1}{2}} \cdot \sqrt{3} = 2^{\frac{3}{2}} \cdot \sqrt{3} \]
Thus, the expression \( \frac{8\sqrt{3}}{2\sqrt{2}} \) simplifies to:
\[ 2^{\frac{3}{2}} \cdot \sqrt{3} \]
If you need just the parts as per your request \( 2^? \):
\[ ? = \frac{3}{2} \]
So,
\[ 2^? \quad ? = \frac{3}{2} \]
If you're looking for the exponent related to the \( \sqrt{3} \) as well:
\(\sqrt{3}\) itself can be written as \(3^{1/2}\) but does not directly fit into the form of \( 2^? \).
In summary, the answer to your question about the exponent of 2 in this context is:
\[ ? = \frac{3}{2} \]
To simplify the expression \(49 \sqrt{7}^{-3}\) and express it in terms of powers of 7, we will use the properties of exponents, including the Quotient Property and Negative Exponent Property.
We recognize that \(49\) can be rewritten as \(7^2\):
\[ 49 = 7^2 \]
Next, we rewrite \(\sqrt{7}\) using rational exponents:
\[ \sqrt{7} = 7^{1/2} \]
Now we rewrite the entire expression:
\[ 49 \sqrt{7}^{-3} = (7^2)(7^{1/2})^{-3} \]
When raising a power to a negative exponent, we can rewrite it as the reciprocal of that power:
\[ (7^{1/2})^{-3} = 7^{-\frac{3}{2}} \]
Now we can combine everything:
\[ 49 \sqrt{7}^{-3} = 7^2 \cdot 7^{-\frac{3}{2}} = 7^{2 - \frac{3}{2}} \]
Now, we simplify the exponent:
\[ 2 - \frac{3}{2} = \frac{4}{2} - \frac{3}{2} = \frac{1}{2} \]
Thus, we have:
\[ 49 \sqrt{7}^{-3} = 7^{\frac{1}{2}} \]
The rational exponent for the new expression is:
\[ ? = \frac{1}{2} \]
So the final answer is:
\[ 7^{?} \quad ? = \frac{1}{2} \]
To determine whether the equation \( 4^3 \cdot (6^4 - \sqrt{3})^2 = 4^6 \) is true or false, we'll simplify the left-hand side.
Starting with the left-hand side:
Calculate \( 4^3 \): \[ 4^3 = 64 \]
Expand \( (6^4 - \sqrt{3})^2 \) using the binomial expansion \( (a - b)^2 = a^2 - 2ab + b^2 \): \[ (6^4 - \sqrt{3})^2 = (6^4)^2 - 2(6^4)(\sqrt{3}) + (\sqrt{3})^2 \]
Putting this together: \[ (6^4 - \sqrt{3})^2 = 6^8 - 2 \cdot 6^4 \cdot \sqrt{3} + 3 \]
The right-hand side is: \[ 4^6 = 4096 \]
Calculating \(64 \cdot (6^8 - 2 \cdot 6^4 \cdot \sqrt{3} + 3)\):
Given that \(6^4\) is \(1296\), \(6^8\) is a much larger number, and thus we can see that:
\[ 4^3 \cdot (6^4 - \sqrt{3})^2 \gg 4^6 \]
Since the left-hand side is much larger than the right-hand side, the equation \( 4^3 \cdot (6^4 - \sqrt{3})^2 = 4^6 \) is false.
Thus, the answer is:
\[ \text{2 (false)} \]
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