Question
1.1.1. \( \left(\frac{3}{4}\right)^{-2}=\frac{(81)^{0}}{3} \) 1.1.2. \( 5^{-3} \cdot 2^{8} \cdot 5^{4} \cdot 2.2^{-2} \) 1.2. \( \quad \) Solve for \( x: \) 1.2.1. \( 2^{-x}=\frac{1}{8} \) 1.2.2. \( \left(3 x^{2}\right)^{3}=1728 \) 1.2.3. \( 2 \times 4^{x}=32 \)
Ask by Burns West. in South Africa
Feb 27,2025
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Tutor-Verified Answer
Answer
1.1.1. \( \left(\frac{3}{4}\right)^{-2} = \frac{16}{9} \), which is not equal to \( \frac{81^0}{3} = \frac{1}{3} \).
1.1.2. The expression simplifies to 640.
1.2.1. \( x = 3 \).
1.2.2. \( x = 2 \) or \( x = -2 \).
1.2.3. \( x = 2 \).
Solution
Below are the detailed steps for each part.
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Problem 1.1.1
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Evaluate
(3/4)^(–2) = ?
and compare it with
(81^0)/3.
Step 1. The negative exponent means we take the reciprocal and square:
(3/4)^(–2) = (4/3)^2.
Step 2. Now square:
(4/3)^2 = 16/9.
Step 3. Evaluate the right-hand side:
81^0 = 1 (since any nonzero number to the 0 power is 1),
so (81^0)/3 = 1/3.
Conclusion:
16/9 ≠ 1/3.
Thus the two expressions are not equal.
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Problem 1.1.2
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Simplify the expression:
5^(–3) · 2^8 · 5^4 · 2 · 2^(–2).
Step 1. Group the factors with the same base.
For base 5:
5^(–3) · 5^4 = 5^(–3+4) = 5^1 = 5.
For base 2:
Notice that 2 · 2^(–2) can be combined as 2^(1–2)=2^(–1).
So for 2 the factors are: 2^8 · 2^(–1)= 2^(8–1)= 2^7 = 128.
Step 2. Multiply the results from the two groups:
5 · 128 = 640.
Thus, the simplified value is 640.
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Problem 1.2.1
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Solve for x:
2^(–x) = 1/8.
Step 1. Express 1/8 as a power of 2.
1/8 = 2^(–3).
Step 2. Since the bases are the same, equate the exponents:
–x = –3.
Step 3. Solve for x:
x = 3.
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Problem 1.2.2
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Solve for x:
(3x²)³ = 1728.
Step 1. Take the cube root of both sides to eliminate the exponent:
3x² = ∛(1728).
Step 2. Recognize that 12³ = 1728, so:
∛(1728) = 12.
Thus, 3x² = 12.
Step 3. Solve for x²:
x² = 12/3 = 4.
Step 4. Take the square root (remembering both positive and negative solutions):
x = 2 or x = –2.
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Problem 1.2.3
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Solve for x:
2 × 4^x = 32.
Step 1. Divide both sides by 2:
4^x = 32/2 = 16.
Step 2. Express 16 as a power of 4.
Since 4² = 16, we have:
4^x = 4^2.
Step 3. Equate the exponents:
x = 2.
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Summary of Answers
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1.1.1. (3/4)^(–2) = 16/9, which is not equal to (81⁰)/3 = 1/3.
1.1.2. The value of the expression is 640.
1.2.1. x = 3.
1.2.2. x = 2 or x = –2.
1.2.3. x = 2.
Answered by UpStudy AI and reviewed by a Professional Tutor
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Simplify this solution The Deep Dive
Let's dive into the exponent world! Understanding the properties of exponents can turn complex problems into simple ones. For instance, remember that \( a^{-n} = \frac{1}{a^{n}} \). This means \( \left(\frac{3}{4}\right)^{-2} \) is the same as \( \left(\frac{4}{3}\right)^{2} \), which opens the door to finding the right answer easily by flipping the fraction and squaring it. When it comes to solving equations like \( 2^{-x}=\frac{1}{8} \), think of it as trying to match bases. Since \( \frac{1}{8} \) can be rewritten as \( 2^{-3} \), set the exponents equal to each other: \( -x = -3 \) leads you straight to \( x=3 \). It’s all about recognizing patterns!
