Answer
- **2.1:** \(\frac{25}{32}\)
- **2.2:** \(x = 3\)
- **2.3:** \((x+y)^{2} = \frac{49}{2}\)
Solution
Simplify the expression by following steps:
- step0: Solution:
\(\frac{\left(3-\sqrt{a}\right)}{\sqrt{2}}+\frac{\left(4+\sqrt{a}\right)}{\sqrt{2}}\)
- step1: Remove the parentheses:
\(\frac{3-\sqrt{a}}{\sqrt{2}}+\frac{4+\sqrt{a}}{\sqrt{2}}\)
- step2: Transform the expression:
\(\frac{3-\sqrt{a}+4+\sqrt{a}}{\sqrt{2}}\)
- step3: Calculate:
\(\frac{7}{\sqrt{2}}\)
- step4: Multiply by the Conjugate:
\(\frac{7\sqrt{2}}{\sqrt{2}\times \sqrt{2}}\)
- step5: Multiply the numbers:
\(\frac{7\sqrt{2}}{2}\)
Solve the equation \( 4^{x}-5*2^{x}-24=0 \).
Solve the equation by following steps:
- step0: Solve for \(x\):
\(4^{x}-5\times 2^{x}-24=0\)
- step1: Factor the expression:
\(\left(2^{x}-8\right)\left(2^{x}+3\right)=0\)
- step2: Separate into possible cases:
\(\begin{align}&2^{x}-8=0\\&2^{x}+3=0\end{align}\)
- step3: Solve the equation:
\(\begin{align}&x=3\\&x \notin \mathbb{R}\end{align}\)
- step4: Find the union:
\(x=3\)
Calculate or simplify the expression \( (2^{m+5}-3*2^{m-2})/(5*2^{m+3}) \).
Simplify the expression by following steps:
- step0: Solution:
\(\frac{\left(2^{m+5}-3\times 2^{m-2}\right)}{\left(5\times 2^{m+3}\right)}\)
- step1: Remove the parentheses:
\(\frac{2^{m+5}-3\times 2^{m-2}}{5\times 2^{m+3}}\)
- step2: Subtract the terms:
\(\frac{125\times 2^{m-2}}{5\times 2^{m+3}}\)
- step3: Reduce the fraction:
\(\frac{25}{2^{5}}\)
- step4: Evaluate the power:
\(\frac{25}{32}\)
Let's go through each part of the problem step by step.
### 2.1 Simplify the expression:
We need to simplify the expression:
\[
\frac{2^{m+5}-3 \cdot 2^{m-2}}{5 \cdot 2^{m+3}}
\]
**Step 1: Factor out common terms in the numerator.**
The numerator can be rewritten as:
\[
2^{m-2}(2^{7} - 3)
\]
This is because \(2^{m+5} = 2^{m-2} \cdot 2^{7}\) and \(3 \cdot 2^{m-2} = 3 \cdot 2^{m-2}\).
**Step 2: Substitute back into the expression.**
Now, substituting this back into the expression gives:
\[
\frac{2^{m-2}(2^{7} - 3)}{5 \cdot 2^{m+3}}
\]
**Step 3: Simplify the fraction.**
We can simplify \(2^{m-2}\) and \(2^{m+3}\):
\[
\frac{2^{m-2}}{2^{m+3}} = \frac{1}{2^{5}} = \frac{1}{32}
\]
Thus, the expression simplifies to:
\[
\frac{2^{7} - 3}{5 \cdot 32}
\]
**Step 4: Calculate \(2^{7} - 3\).**
Calculating \(2^{7} = 128\):
\[
128 - 3 = 125
\]
So, we have:
\[
\frac{125}{160} = \frac{25}{32}
\]
The fully simplified expression is:
\[
\frac{25}{32}
\]
### 2.2 Solve for \( x \):
We need to solve the equation:
\[
4^{x}-5 \cdot 2^{x}-24=0
\]
**Step 1: Rewrite \(4^{x}\) in terms of \(2^{x}\).**
Since \(4^{x} = (2^{2})^{x} = 2^{2x}\), we can rewrite the equation as:
\[
2^{2x} - 5 \cdot 2^{x} - 24 = 0
\]
**Step 2: Let \(y = 2^{x}\).**
This transforms the equation into a quadratic form:
\[
y^{2} - 5y - 24 = 0
\]
**Step 3: Factor or use the quadratic formula.**
Using the quadratic formula \(y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\):
Here, \(a = 1\), \(b = -5\), and \(c = -24\):
\[
y = \frac{5 \pm \sqrt{(-5)^{2} - 4 \cdot 1 \cdot (-24)}}{2 \cdot 1}
\]
Calculating the discriminant:
\[
25 + 96 = 121
\]
So,
\[
y = \frac{5 \pm 11}{2}
\]
Calculating the two possible values for \(y\):
1. \(y = \frac{16}{2} = 8\)
2. \(y = \frac{-6}{2} = -3\) (not valid since \(y = 2^{x}\) must be positive)
Thus, we have:
\[
2^{x} = 8 \implies x = 3
\]
### 2.3 Determine the value of \( (x+y)^{2} \):
Given:
\[
x = \frac{3 - \sqrt{a}}{\sqrt{2}} \quad \text{and} \quad y = \frac{4 + \sqrt{a}}{\sqrt{2}}
\]
**Step 1: Calculate \(x + y\).**
\[
x + y = \frac{3 - \sqrt{a}}{\sqrt{2}} + \frac{4 + \sqrt{a}}{\sqrt{2}} = \frac{(3 - \sqrt{a}) + (4 + \sqrt{a})}{\sqrt{2}} = \frac{7}{\sqrt{2}}
\]
**Step 2: Calculate \((x+y)^{2}\).**
\[
(x+y)^{2} = \left(\frac{7}{\sqrt{2}}\right)^{2} = \frac{49}{2}
\]
### Final Answers:
- **2.1:** \(\frac{25}{32}\)
- **2.2:** \(x = 3\)
- **2.3:** \((x+y)^{2} = \frac{49}{2}\)
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