Let's solve each of the individual questions one by one:
Question 1
Find the difference:
\[
-7\sqrt{8} - (-6\sqrt{8})
\]
This simplifies to:
\[
-7\sqrt{8} + 6\sqrt{8} = (-7 + 6)\sqrt{8} = -1\sqrt{8} = -\sqrt{8}
\]
Answer: \(-\sqrt{8}\)
Reasoning: This is calculated by combining like terms under the square root.
Question 2
Find the difference:
\[
28\sqrt{363} - 3\sqrt{27}
\]
First, simplify \( \sqrt{27} = 3\sqrt{3} \):
\[
28\sqrt{363} - 3(3\sqrt{3}) = 28\sqrt{363} - 9\sqrt{3}
\]
Answer: \(28\sqrt{363} - 9\sqrt{3}\)
Reasoning: The operation simplifies the square root and results in two distinct terms.
Question 3
Find the quotient:
\[
\frac{7\sqrt{420}}{\sqrt{14}}
\]
This can be rewritten as:
\[
7\cdot\frac{\sqrt{420}}{\sqrt{14}} = 7\sqrt{\frac{420}{14}} = 7\sqrt{30}
\]
Answer: \(7\sqrt{30}\)
Reasoning: Simplifying under a single square root gives a clearer answer.
Question 4
Evaluate the expression:
\[
\left(\frac{11^{3/8}}{11^{5/8}}\right)^8 = (11^{(3/8 - 5/8)})^8 = (11^{-2/8})^8 = 11^{-2} = \frac{1}{11^2} = \frac{1}{121}
\]
Answer: \(\frac{1}{121}\)
Reasoning: Using properties of exponents to simplify.
Question 5
Generate an equivalent expression:
\[
\left(\frac{32^3}{32^{1/2}}\right)^{-4} = \left(32^{3 - 1/2}\right)^{-4} = \left(32^{2.5}\right)^{-4} = 32^{-10}
\]
Answer: \(32^{-10}\)
Reasoning: The rules of exponents allow us to combine terms and simplify.
Question 6
Which expression is equivalent to:
\(\sqrt{180}\)
First, simplify:
\(\sqrt{180} = \sqrt{36 \cdot 5} = 6\sqrt{5}\)
None of the provided options are equivalent to \(\sqrt{180}\), so: \(c. 6\sqrt{5}\).
Answer: \(c\)
Reasoning: Recognizing perfect squares and simplifying the root.
Question 7
Evaluate the expression:
\[
100^{3/2} = (10^2)^{3/2} = 10^{3} = 1000
\]
Answer: \(1000\)
Reasoning: Here, properties of exponents simplify the power.
Question 8
Simplify:
\(\sqrt{1764}\)
Finding the square root:
\(\sqrt{1764} = 42\).
Answer: \(42\)
Reasoning: Recognized as a perfect square.
Question 9
Find the product:
\(\sqrt{28} \cdot 7\sqrt{10} = 7\cdot\sqrt{28 \cdot 10} = 7\sqrt{280}\)
Answer: \(7\sqrt{280}\)
Reasoning: Combining terms under a radical.
Question 10
Find the sum:
\(5\sqrt[3]{48} + 18\sqrt[3]{-384}\).
First find each separately.
\(\sqrt[3]{48} = 4\sqrt[3]{3}\) and \(\sqrt[3]{-384} = -4\sqrt[3]{6}\)
The total becomes:
\(20\sqrt[3]{3} - 72\sqrt[3]{6}\).
Answer: \(20\sqrt[3]{3} - 72\sqrt[3]{6}\)
Reasoning: Combining terms with respective radicals.
Question 11
Convert to radical and simplify:
\[
9^{6/3} = 9^2 = 81
\]
Answer: \(81\)
Reasoning: Recognizing integer powers simplifies the expression.
Question 12
Find the sum:
\(10\sqrt{12} + 9\sqrt{12} = (10 + 9)\sqrt{12} = 19\sqrt{12}\)
Answer: \(19\sqrt{12}\)
Reasoning: Combining like terms in square roots.
Question 13
Identify the equivalent expression:
\[
\frac{x^{-10}y^{-4}}{y^{10}x^0} = \frac{x^{-10}}{x^0}\cdot\frac{1}{y^{10}}y^{-4} = \frac{y^{-4}}{y^{10}}x^{-10} = \frac{1}{x^{10}y^{14}}
\]
Answer: \(a.\)
Reasoning: Simplifying using the rules of exponents.
Question 14
Simplify:
\(-4\sqrt{729} = -4 \cdot 27 = -108\)
Answer: \(-108\)
Reasoning: Finding the perfect square root.
Question 15
Select all properties:
a. Product of a Power
d. Negative Exponent
Answer: a, d.
Reasoning: The properties would help in simplifying the expression.
Question 16
Find the sum:
\(-6\sqrt[3]{12} -19\sqrt[3]{12} = -25\sqrt[3]{12}\)
Answer: \(-25\sqrt[3]{12}\)
Reasoning: Combining like terms.
Question 17
Rewrite in exponential form:
\(\sqrt[7]{40^{11}} = 40^{\frac{11}{7}}\)
Answer: \(40^{\frac{11}{7}}\)
Reasoning: Converting radicals to exponents.
Question 18
Write an equivalent expression:
\[
\frac{7^{-9} \cdot 5^{-8} \cdot 5^7}{5^{-3} \cdot 7^8 \cdot 7^{-3}} = \frac{7^{-9}}{7^{11}} \cdot 5^{-1}
= 7^{-20} \cdot 5^{-1} = \frac{1}{7^{20} \cdot 5}
\]
Answer: \(\frac{1}{7^{20} \cdot 5}\)
Reasoning: Using properties of negative exponents for simplification.
Question 19
Rewrite as radical:
\[
31^{\frac{9}{5}} = \sqrt[5]{31^9}
\]
Answer: \(\sqrt[5]{31^9}\)
Reasoning: Converting the power to radical form.
Question 20
Generate equivalent expression:
\[
(2 x^{-6} y^{7})^3 (x^{-6} y^{3}) = 2^3 x^{-18} y^{21} x^{-6} y^{3} = 8 x^{-24} y^{24}
\]
Answer: \(8x^{-24}y^{24}\)
Reasoning: Combine and simplify using exponent rules.
Question 21
Finding x:
\[
9^{14} = (9^2 / 9^9)^x = 9^{2-9x}
\]
Setting equal exponents:
\(14 = 2 - 9x\) leads to \(9x = 2 - 14\) giving \(x = 4/9\)
Answer: \(x = \frac{4}{9}\)
Reasoning: Using exponent properties to solve.
Question 22
Simplify:
\[
10\sqrt[3]{-108} = 10 \cdot (-3) \cdot \sqrt[3]{4} = -30\sqrt[3]{4}
\]
Answer: \(-30\sqrt[3]{4}\)
Reasoning: Breaking down cube roots and numbers.
Question 23
Which expression is equivalent:
\((-7)^{-3} = \left(\frac{-1}{7}\right)^3\)
b. \(\left(\frac{1}{7}\right)^3\) is incorrect.
c is the proper choice.
Answer: \(c.\)
Reasoning: Evaluating negative exponents leads to the correct simplification.
Question 24
Single exponent expression:
\[
(-\frac{3}{7})^{-3} \cdot (-7)^{-3} = (-1)^{-3} \cdot 3^{-3} \cdot 7^{-3} \cdot (-7)^{-3} = 7^{-6}
\]
Answer: \((-1)^{-3} \cdot (3/7)^{-3}\)
Reasoning: Applying properties of multiplication with exponents.
Question 25
Evaluate:
\[
(8^{1/9} \cdot 8^{1/3})^{-3} = 8^{(1/9 + 1/3)(-3)} = 8^{-8/9} = \frac{1}{8^{8/9}}
\]
Answer: \(\frac{1}{8^{8/9}}\)
Reasoning: Consolidating exponents accurately.
Question 26
Generate equivalent expression:
\[
(\frac{2x^{-10} y^{-1}}{4x^{-10}})^{1/3} = (\frac{2}{4} y^{-1})^{1/3} = (\frac{1}{2} y^{-1})^{1/3} = \frac{1}{2^{1/3}y^{1/3}}
\]
Answer: \(\frac{1}{2^{1/3}y^{1/3}}\)
Reasoning: Applying fraction properties to the expression.
Question 27
Which represents:
12\(\sqrt{29} - (-\sqrt{29}) = (12 + 1)\sqrt{29} = 13\sqrt{29}\)
Answer: \(c.\)
Reasoning: Combining like terms effectively.
If you have more queries or need further clarification, feel free to ask!