1.
Write the product of 5^{2} x 5^{4} as a single power.
Correct Answer
B. 5^{6}
Explanation
To find the product of two numbers as a single power, you add the exponents. In this case, both numbers have a base of 2 and different exponents. 52 can be written as 2^2 and 54 can be written as 2^3. Adding the exponents gives us 2^5, which is equal to 32. Therefore, the product of 52 x 54 as a single power is 32.
2.
Evaluate: (2+3)^{2} – (3-5)^{3}
Correct Answer
D. 33
Explanation
The given expression can be simplified using the order of operations. First, we evaluate the parentheses: (2+3) equals 5 and (3-5) equals -2. Then, we calculate the exponents: 5^2 equals 25 and -2^3 equals -8. Finally, we subtract -8 from 25, resulting in 33.
3.
Write [(-7) x 2]^{5} as a product of powers.
Correct Answer
A. (-7)^{5} x 2^{5}
Explanation
The given expression [(-7) x 2]5 can be simplified by first performing the multiplication (-7) x 2, which equals -14. Then, raising -14 to the power of 5 gives the final result of (-7)5. Therefore, the expression can be written as (-7)5 x 25.
4.
Evaluate: - (10^{0})^{7}
Correct Answer
D. -1
Explanation
When evaluating the expression - (100)7, we start by multiplying 100 by 7, which gives us 700. Then, since there is a negative sign in front of the expression, we change the sign of the result to -700. Therefore, the correct answer is -700.
5.
Simplify the following:
x^{2}x^{3}
Correct Answer
D. X^{5}
Explanation
The given expression x2x3 can be simplified by adding the exponents of x, which gives us x5. Therefore, the answer is x5.
6.
Write the base of −(−6)^{5}
Correct Answer
B. -6
Explanation
The base of -(-6)5 is -6.
7.
Write the quotient of 6^{8} / 6^{4} as a single power.
Correct Answer
A. 6^{4}
Explanation
To find the quotient of 68 divided by 64, we divide the numerator (68) by the denominator (64). The quotient is 1 with a remainder of 4. Therefore, the fraction 68/64 can be simplified to 1 and 4/64. Since 4/64 can be simplified further to 1/16, the final simplified form of 68/64 is 1 + 1/16.
8.
Find n so that the equation is true: (5^{3})^{n} = 5^{18}
Correct Answer
A. N=6
Explanation
To find the value of n that makes the equation true, we need to solve the equation (53)n = 518. By evaluating the powers of 53, we find that 53^6 = 113,379,904, which is not equal to 518. Therefore, n=6 is not the correct answer.
9.
Using the rules of exponents, simplify the following expression. Your answer should be in exponential form. 7^{3} . 5^{6} /7 . 5^{2}
Correct Answer
C. 7^{2} . 5^{4}
Explanation
The expression can be simplified using the rule of exponents that states "a^m . a^n = a^(m+n)". In this case, we have 73 . 56 / 7 . 52. We can simplify this by adding the exponents of 7 and 5, which gives us 72. Then, we can multiply the exponents of 3 and 2, which gives us 54. Therefore, the simplified expression in exponential form is 72 . 54.
10.
Evaluate: [(-z)^{5}]^{4}
Correct Answer
C. Z^{20}
Explanation
The expression (-z)5 means that the value of z is multiplied by -1 five times. So, when we evaluate [(-z)5]4, we are multiplying the result of (-z)5 by 4. Since (-z)5 is equal to -z^5, the expression becomes -z^5 * 4. This can be simplified to -4z^5. Therefore, the correct answer is -z20.
11.
Find the value of the polynomial when x = - 3;
2x^{3}- 5x^{2} +6x - 7
Correct Answer
C. -34
Explanation
When we substitute x = -3 into the polynomial 2x^3 - 5x^2 + 6x - 7, we get:
2(-3)^3 - 5(-3)^2 + 6(-3) - 7
= 2(-27) - 5(9) - 18 - 7
= -54 - 45 - 18 - 7
= -124
Therefore, the value of the polynomial when x = -3 is -124.
12.
Evaluate: 3x^{4} + xy - 5x - y for x = 2, y = 3
Correct Answer
D. 41
Explanation
The given expression is 3x4 + xy - 5x - y. Substituting x = 2 and y = 3 into the expression, we get 3(2^4) + 2(3) - 5(2) - 3. Simplifying this expression, we have 3(16) + 6 - 10 - 3 = 48 + 6 - 10 - 3 = 41. Therefore, the correct answer is 41.
13.
Solve: - 2(x+1)^{2} + 2 for x = 1
Correct Answer
B. -6
Explanation
To solve the equation, we substitute x = 1 into the equation and simplify. Plugging in x = 1, we have: -2(1+1)2 + 2. Simplifying further, we get: -2(2)2 + 2 = -2(4) + 2 = -8 + 2 = -6. Therefore, the correct answer is -6.
14.
/2x+3/ - 7 for x = -5
Correct Answer
C. 0
Explanation
The expression |2x+3| - 7 is being evaluated for x = -5. When x = -5, the expression becomes |2(-5)+3| - 7. Simplifying further, we have |-10+3| - 7, which becomes |-7| - 7. The absolute value of -7 is 7, so the expression becomes 7 - 7, which equals 0. Therefore, the correct answer is 0.
15.
Evaluate: 3^{-4}
Correct Answer
B. 1/81
Explanation
The expression "3-4" simplifies to -1. When we divide -1 by 1/3, we can rewrite it as -1 * 3/1, which equals -3. Dividing -3 by 1/81 is the same as multiplying -3 by 81/1, resulting in -243. Therefore, the correct answer is 1/81.
16.
Solve: (x^{-9}) (x^{6}) = ?
Correct Answer
A. 1/x^{3}
Explanation
The given expression (x-9)(x6) can be simplified by multiplying the terms inside the parentheses. When we multiply x by x6, we get x^7. Similarly, when we multiply -9 by x6, we get -9x^6. Therefore, the simplified expression is x^7 - 9x^6. However, this does not match any of the answer choices provided. Therefore, the correct answer is not available.
17.
Evaluate: x^{4} - 5x^{2} - 2x + 1 for x = -2
Correct Answer
C. -7
Explanation
To evaluate the expression x^4 - 5x^2 - 2x + 1 for x = -2, we substitute -2 into the expression. Replacing x with -2, we get (-2)^4 - 5(-2)^2 - 2(-2) + 1. Simplifying this expression, we have 16 - 5(4) + 4 + 1 = 16 - 20 + 4 + 1 = 1 - 20 + 4 + 1 = -19 + 5 = -14. Therefore, the correct answer is -14.
18.
Evaluate: (2xy^{2}/z^{4})^{3} =?
Correct Answer
B. 8x^{3}y^{6}/z^{12}
Explanation
The given expression is (2xy^2/z^4)^3. To simplify this expression, we raise each term inside the parentheses to the power of 3. Therefore, 2^3 = 8, (x)^3 = x^3, (y^2)^3 = y^6, and (z^4)^3 = z^12. Combining these terms, we get the simplified expression 8x^3y^6/z^12, which matches the given answer.
19.
Simplify the following: (b^{5}) (b^{2}) (b^{7}) = ?
Correct Answer
A. B^{14}
Explanation
The given expression involves multiplying three terms with the same base b. When multiplying terms with the same base, we add their exponents. In this case, the exponents are 5, 2, and 7. Adding these exponents gives us 14. Therefore, the simplified expression is b14.
20.
In the given: (xy)^{4} = x^{4}y^{4}. What rules of exponents is being applied?
Correct Answer
C. Product Rule
Explanation
The product rule of exponents is being applied in this case. The product rule states that when raising a product of two variables to a power, each variable is raised to that power individually. In the given equation, (xy)^4, both x and y are being raised to the power of 4 individually, resulting in x^4 * y^4.
21.
Find -2x^{2} when x = -1
Correct Answer
B. -2
Explanation
When we substitute x = -1 into the expression -2x^2, we get -2(-1)^2 = -2(1) = -2. Therefore, the value of -2x^2 when x = -1 is -2.
22.
Solve: 8r/r + 3r -1 where r = 3
Correct Answer
B. 16
Explanation
The given expression is 8r/r + 3r - 1. By substituting r = 3 into the expression, we get 8(3)/3 + 3(3) - 1. Simplifying this further, we have 24/3 + 9 - 1, which equals 8 + 9 - 1. Finally, adding these numbers together, we get 16.
23.
Simplify: (-2p^{3}q^{5})^{2} = ?
Correct Answer
A. 4p^{6}q^{10}
Explanation
The given expression (-2p3q5)2 can be simplified by raising each term inside the parentheses to the power of 2. This results in (-2)2 * p3*2 * q5*2 = 4p6q10.
24.
Evaluate: (-3)^{0} = 1. What rule of exponent is being applied?
Correct Answer
C. Zero Exponent
Explanation
The correct answer is Zero Exponent. The zero exponent rule states that any number raised to the power of zero is equal to 1. In this case, (-3) raised to the power of zero is equal to 1.
25.
If you are raising a Power to an exponent, you multiply the exponents. This is the definition of what laws of exponents?
Correct Answer
C. Power of a Power
Explanation
The explanation for the correct answer, Power of a Power, is that when raising a power to an exponent, you multiply the exponents. This means that if you have a base raised to an exponent, and that entire expression is raised to another exponent, you can simplify it by multiplying the exponents together. This is a fundamental rule in the laws of exponents and is known as the Power of a Power rule.
26.
Evaluate: (3^{9}/3^{5})^{-2}
Correct Answer
D. 1/3^{8}
Explanation
The expression (39/35)-2 can be simplified by first dividing 39 by 35, which equals 1.114. Then subtracting 2 from 1.114 gives us -0.886. Therefore, the correct answer is -0.886.
27.
Evaluate the given algebraic expression: 2x^{3}- 5x^{2} + 6x - 7
Correct Answer
A. 110
Explanation
The given algebraic expression is 2x^3 - 5x^2 + 6x - 7. To evaluate this expression, we substitute the value of x with 5. So, when x = 5, the expression becomes 2(5)^3 - 5(5)^2 + 6(5) - 7. Simplifying this expression, we get 250 - 125 + 30 - 7 = 148. Since the given answer options do not match with the evaluated value, the correct answer is not available.
28.
Simplify: (3^{2}) (3^{4}) = ?
Correct Answer
C. 729
Explanation
To simplify the expression (32)(34), we can multiply the numbers together. 32 multiplied by 34 equals 1,088. However, since the question asks for a simplified answer, we need to find the square root of 1,088. The square root of 1,088 is equal to 32, so the simplified answer is 32.
29.
Evaluate & Simplify: (-2)^{4} (-2)^{5} = ?
Correct Answer
D. -512
Explanation
To evaluate this expression, we need to apply the exponent rules. The rule states that when multiplying two numbers with the same base, we add the exponents. In this case, we have (-2) raised to the power of 4 multiplied by (-2) raised to the power of 5. This can be simplified as (-2)^(4+5), which is equal to (-2)^9. Since the base is negative, the result will also be negative. Therefore, the correct answer is -512.
30.
What is the value of x^{0} = ?
Correct Answer
C. 1
Explanation
The value of x0 is 1.