Use the Product Rule of Exponents to simplify the expression 24 ⋅22 .
To simplify the expression 24 ⋅22 using the product rule of exponents, we add the exponents together:
24 ⋅ 22 = 2^4 ⋅ 2^2 = 2^(4+2) = 2^6
So, 24 ⋅22 simplifies to 2^6.
Question
Use the Product Rule of Exponents to simplify the expression 3
6^8 ⋅3^3 .
To simplify the expression 6^8 ⋅ 3^3 using the product rule of exponents, we add the exponents together:
6^8 ⋅ 3^3 = (2^3)^8 ⋅ 3^3 = 2^(3*8) ⋅ 3^3 = 2^24 ⋅ 3^3
So, 6^8 ⋅ 3^3 simplifies to 2^24 ⋅ 3^3.
Use the Product Rule of Exponents to simplify the expression 75 ⋅72
To simplify the expression 75 ⋅ 72 using the product rule of exponents, we need to rewrite the numbers in terms of their prime factorization:
75 = 3^1 ⋅ 5^2
72 = 2^3 ⋅ 3^2
Now we can use the product rule of exponents to simplify:
75 ⋅ 72 = (3^1 ⋅ 5^2) ⋅ (2^3 ⋅ 3^2)
Since the bases are different (3 and 2), we cannot simplify further using the product rule of exponents. Therefore, the expression 75 ⋅ 72 cannot be further simplified.
Which expression is correctly developed to use the Product Rule of Exponents?(1 point)
Responses
(52)9
left parenthesis 5 squared right parenthesis superscript 9 baseline
62⋅73
6 squared times 7 cubed
108⋅108
10 superscript 8 baseline times 10 superscript 8 baseline
327
The expression that is correctly developed to use the Product Rule of Exponents is 62⋅73 (6 squared times 7 cubed).
Which of the following correctly used the Product Rule of Exponents to simplify the expression 1008⋅1007?(1 point)
Responses
10015
100 superscript 15 baseline
10056
100 superscript 56 baseline
20015
200 superscript 15 baseline
10,00015
To simplify the expression 1008⋅1007 using the Product Rule of Exponents, we add the exponents together:
1008 = 2^4 ⋅ 3^2 ⋅ 7^1
1007 = 2^0 ⋅ 3^0 ⋅ 7^1
By multiplying the bases together separately, we can use the Product Rule of Exponents:
1008⋅1007 = 2^4 ⋅ 2^0 ⋅ 3^2 ⋅ 3^0 ⋅ 7^1 ⋅ 7^1 = 2^(4+0) ⋅ 3^(2+0) ⋅ 7^(1+1) = 2^4 ⋅ 3^2 ⋅ 7^2
Therefore, the correct expression that uses the Product Rule of Exponents to simplify 1008⋅1007 is 20015 (200 superscript 15 baseline).
Find the numerical equivalent of 102 ⋅103(1 point)
Responses
100,000
100,000
105
10 superscript 5 baseline
10
10
1,100
To find the numerical equivalent of 102 ⋅ 103, we simply multiply the two numbers together:
102 ⋅ 103 = 100 ⋅ 1,000 = 100,000
Therefore, the numerical equivalent of 102 ⋅ 103 is 100,000.
Find the numerical equivalent of 25 ⋅2−2(1 point)
Responses
128
128
8
8
2
2
16
To find the numerical equivalent of 25 ⋅ 2^(-2), we evaluate the expression by simplifying the exponent:
2^(-2) is the same as 1 / 2^2, which equals 1 / 4 or 1/4.
Now we multiply 25 by 1/4:
25 * 1/4 = 25/4
Therefore, the numerical equivalent of 25 ⋅ 2^(-2) is 25/4.
Find the numerical equivalent of 49⋅4−7(1 point)
Responses
416
4 superscript 16 baseline
16
16
42
4 squared
4
To find the numerical equivalent of 49 ⋅ 4^(-7), we evaluate the expression by simplifying the exponent:
4^(-7) is the same as 1 / 4^7, which equals 1 / 16384 or 1/16384.
Now we multiply 49 by 1/16384:
49 * 1/16384 ≈ 0.00299
Therefore, the numerical equivalent of 49 ⋅ 4^(-7) is approximately 0.00299.
Simplify this expression using the Quotient Rule of Integer Exponents: t400t100.(1 point)
Responses
300
300
t300
t superscript 300 baseline
t4
t superscript 4 baseline
t500
To simplify the expression t^400 / t^100 using the Quotient Rule of Integer Exponents, we subtract the exponents:
t^400 / t^100 = t^(400 - 100) = t^300
Therefore, the simplified expression using the Quotient Rule of Integer Exponents is t^300.
Simplify a67b34.(1 point)
Responses
23
23
a67b34
Start Fraction a superscript 67 baseline over b superscript 34 baseline End Fraction
ab33
a b superscript 33 baseline
a33
The expression a^67b^34 cannot be simplified further or written in another form. Therefore, the simplified expression is still a^67b^34.
Apply the Quotient Rule of Integer Exponents to simplify 2365÷2332.(1 point)
Responses
2397
23 superscript 97 baseline
97
97
133
1 superscript 33 baseline
2333
To apply the Quotient Rule of Integer Exponents to simplify 2365 ÷ 2332, we subtract the exponents:
2365 ÷ 2332 = 2^(365 - 332) = 2^33
Therefore, the simplified expression using the Quotient Rule of Integer Exponents is 2^33.
Simplify this expression using the Quotient Rule of Integer Exponents: 1253÷753.(1 point)
Responses
12106
12 superscript 106 baseline
553
5 superscript 53 baseline
1253÷753
12 superscript 53 baseline divided by 7 superscript 53 baseline
5106
To simplify the expression 1253 ÷ 753 using the Quotient Rule of Integer Exponents, we subtract the exponents:
1253 ÷ 753 = 5^(253 - 53) = 5^200
Therefore, the simplified expression using the Quotient Rule of Integer Exponents is 5^200.
Which are the correct steps to follow to simplify an exponential expression using the Quotient Rule of Integer Exponents?(1 point)
Responses
Check if the bases are different; if they are, subtract the exponents of the numerator and denominator; and write the resulting expression in exponential form.
Check if the bases are different; if they are, subtract the exponents of the numerator and denominator; and write the resulting expression in exponential form.
Check if the bases are the same; if they are, subtract the exponents of the numerator and denominator (leave as-is otherwise); and write the resulting expression in exponential form.
Check if the bases are the same; if they are, subtract the exponents of the numerator and denominator (leave as-is otherwise); and write the resulting expression in exponential form.
Check if the bases are the same; if they are, add the exponents of the numerator and denominator (leave as-is otherwise); and write the resulting expression in exponential form.
Check if the bases are the same; if they are, add the exponents of the numerator and denominator (leave as-is otherwise); and write the resulting expression in exponential form.
Subtract the exponents of the numerator and denominator; and write the resulting expression in exponential form.
The correct steps to follow to simplify an exponential expression using the Quotient Rule of Integer Exponents are:
1. Check if the bases are the same. If they are, subtract the exponents of the numerator and denominator (leave as-is otherwise).
2. Write the resulting expression in exponential form.
So the correct response is: Check if the bases are the same; if they are, subtract the exponents of the numerator and denominator (leave as-is otherwise); and write the resulting expression in exponential form.
Which of the following is developed to result in the answer of 1?(1 point)
Responses
50+90
5 superscript 0 baseline plus 9 superscript 0 baseline
13−(7.42)0
13 minus left parenthesis 7.42 right parenthesis superscript 0 baseline
(−2.97)0
left parenthesis negative 2.97 right parenthesis superscript 0 baseline
3(80)
The expression that is developed to result in the answer of 1 is (−2.97)^0. According to the rule that any non-zero base raised to the power of 0 is equal to 1, the expression (-2.97)^0 is equal to 1.
Which of the following shows the correct process of solving −30+4.5?(1 point)
Responses
−30+4.5=(−3)0+4.5=1+4.5=5.5
negative 3 superscript 0 baseline plus 4.5 equals left parenthesis negative 3 right parenthesis superscript 0 baseline plus 4.5 equals 1 plus 4.5 equals 5.5
−30+4.5=−1⋅30+4.5=−1⋅1+4.5=−1+4.5=3.5
negative 3 superscript 0 baseline plus 4.5 equals negative 1 times 3 superscript 0 baseline plus 4.5 equals negative 1 dot 1 plus 4.5 equals negative 1 plus 4.5 equals 3.5
−30+4.5=−3⋅0+4.5=0+4.5=4.5
negative 3 superscript 0 baseline plus 4.5 equals negative 3 times 0 plus 4.5 equals 0 plus 4.5 equals 4.5
−30+4.5=−3+4.5=1.5
The correct process of solving −30 + 4.5 is:
−30 + 4.5 = −3 + 4.5 = 1.5
Therefore, the correct response is −30 + 4.5 = −3 + 4.5 = 1.5.
Which of the following is an equivalent expression to 7(−5.3)0+4⋅9 when applying the Zero Power Rule? (1 point)
Responses
75.30+36
Start Fraction 7 over 5.3 superscript 0 baseline End Fraction plus 36
70+4⋅9
Start Fraction 7 over 0 End Fraction plus 4 times 9
71+4⋅9
Start Fraction 7 over 1 End Fraction plus 4 times 9
75.3+4⋅9
When applying the Zero Power Rule, any non-zero base raised to the power of 0 is equal to 1. Therefore, an equivalent expression to 7(−5.3)^0 + 4⋅9 would be:
7(−5.3)^0 + 4⋅9 = 7(1) + 4⋅9
Since anything multiplied by 1 does not change the value, this simplifies to:
7 + 4⋅9
So, the correct response is: 7 + 4⋅9.
Which of the following is an equivalent expression to 12(9−70)+(−29)0? (1 point)
Responses
12(9−1)−1
Start Fraction 1 over 2 End Fraction left parenthesis 9 minus 1 right parenthesis minus 1
12(2)+1
Start Fraction 1 over 2 End Fraction left parenthesis 2 right parenthesis plus 1
12(9−0)+0
Start Fraction 1 over 2 End Fraction left parenthesis 9 minus 0 right parenthesis plus 0
12(9−1)+1
To find an equivalent expression to 12(9−70)+(−29)^0, we can simplify it using the Zero Power Rule which states that any non-zero base raised to the power of 0 is equal to 1:
12(9−70)+(−29)^0 = 12(-61) + 1
Now, we can simplify further:
12(-61) + 1 = -732 + 1 = -731
Therefore, the equivalent expression to 12(9−70)+(−29)^0 is -731.
Which of the following shows the correct process of solving −4(3+120)+150 by generating an equivalent expression using the Zero Power Rule? (1 point)
Responses
−4(3+120)+150 =−4(15)+15=−60+15=−45
negative 4 left parenthesis 3 plus 12 superscript 0 baseline right parenthesis plus 15 superscript 0 baseline equals negative 4 left parenthesis 15 right parenthesis plus 15 equals negative 60 plus 15 equals negative 45
−4(3+120)+150=−4(3+1)+1=−4⋅4+1=−4⋅5=−20
negative 4 left parenthesis 3 plus 12 superscript 0 baseline right parenthesis plus 15 superscript 0 baseline equals negative 4 left parenthesis 3 plus 1 right parenthesis plus 1 equals negative 4 times 4 plus 1 equals negative 4 times 5 equals negative 20
−4(3+120)+150=−4(3+1)+1=−4⋅4+1=−16+1=−15
negative 4 left parenthesis 3 plus 12 superscript 0 baseline right parenthesis plus 15 superscript 0 baseline equals negative 4 left parenthesis 3 plus 1 right parenthesis plus 1 equals negative 4 times 4 plus 1 equals negative 16 plus 1 equals negative 15
−4(3+120)+150=−4(15)+1=−60+1=−59
To solve the expression −4(3+120)+150 and generate an equivalent expression using the Zero Power Rule, we simplify the expression inside the parentheses, perform the multiplication, and then perform the addition:
−4(3+120)+150 = −4(123) + 150 = -492 + 150 = -342
Therefore, the correct response is: −4(3+120)+150 = -4(123) + 150 = -342.
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Question
Which expression is correctly developed to use the Power Rule of Exponents?(1 point)
Responses
(78)⋅(710)
left parenthesis 7 superscript 8 baseline right parenthesis times left parenthesis 7 superscript 10 baseline right parenthesis
(−5)0
left parenthesis negative 5 right parenthesis superscript 0 baseline
129
12 superscript 9 baseline
(74)8
The expression that is correctly developed to use the Power Rule of Exponents is (7^8) ⋅ (7^10) = (7^(8+10)) = 7^18.
Which of the following correctly used the Power Rule of Exponents in the expression (109)7?(1 point)
Responses
907
90 superscript 7 baseline
102
10 superscript 2 baseline
1063
10 superscript 63 baseline
1016
To correctly use the Power Rule of Exponents in the expression (10^9)^7, we multiply the exponents:
(10^9)^7 = 10^(9*7) = 10^63
Therefore, the correct response is 10^63 (10 superscript 63 baseline).
If the expression 410 was simplified using the Power Rule of Exponents, what is an equivalent expanded expression? (1 point)
Responses
(45)2
left parenthesis 4 superscript 5 baseline right parenthesis squared
(412)2
left parenthesis 4 superscript 12 baseline right parenthesis squared
(42)8
left parenthesis 4 squared right parenthesis superscript 8 baseline
4⋅4⋅4⋅4⋅4⋅4⋅4
If the expression 410 was simplified using the Power Rule of Exponents, an equivalent expanded expression would be 4⋅4⋅4⋅4⋅4⋅4⋅4, which is the result of multiplying 4 by itself 10 times.
Use the Power Rule of Exponents to create an expression equivalent to (156)2.(1 point)
Responses
1512
15 superscript 12 baseline
153
15 cubed
158
15 superscript 8 baseline
154
To create an expression equivalent to (15^6)^2 using the Power Rule of Exponents, we multiply the exponents:
(15^6)^2 = 15^(6*2) = 15^12
Therefore, the equivalent expression is 15^12 (15 to the power of 12).
What exponent would make the two expressions equivalent: (33)x=324?(1 point)
Responses
8
8
27
27
21
21
15