Solved on Feb 22, 2024

Use the product rule to simplify and find the values of $4^{5} \times 4^{2}$ , $(-2)^{2} \times(-2)^{4} \times(-2)$ , $2^{3} \times 2^{2} \times 2^{2}$ , and $\left(\frac{3}{4}\right)^{3} \times\left(\frac{3}{4}\right)^{3}$ .

STEP 1

Assumptions
1. We will use the product rule of exponents which states that $a^m \times a^n = a^{m+n}$ .
2. We will simplify each expression using the product rule.
3. After simplifying, we will calculate the value of each expression.

STEP 2

Simplify the expression $4^{5} \times 4^{2}$ using the product rule.
$4^{5} \times 4^{2} = 4^{5+2}$

STEP 3

Calculate the exponent.
$4^{5+2} = 4^{7}$

STEP 4

Find the value of $4^{7}$ .
$4^{7} = 4 \times 4 \times 4 \times 4 \times 4 \times 4 \times 4$

STEP 5

Calculate the value.
$4^{7} = 16384$

STEP 6

Simplify the expression $(-2)^{2} \times(-2)^{4} \times(-2)$ using the product rule.
$(-2)^{2} \times(-2)^{4} \times(-2) = (-2)^{2+4+1}$

STEP 7

Calculate the exponent.
$(-2)^{2+4+1} = (-2)^{7}$

STEP 8

Find the value of $(-2)^{7}$ .
$(-2)^{7} = -2 \times -2 \times -2 \times -2 \times -2 \times -2 \times -2$

STEP 9

Calculate the value.
$(-2)^{7} = -128$

STEP 10

Simplify the expression $2^{3} \times 2^{2} \times 2^{2}$ using the product rule.
$2^{3} \times 2^{2} \times 2^{2} = 2^{3+2+2}$

STEP 11

Calculate the exponent.
$2^{3+2+2} = 2^{7}$

STEP 12

Find the value of $2^{7}$ .
$2^{7} = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2$

STEP 13

Calculate the value.
$2^{7} = 128$

STEP 14

Simplify the expression $\left(\frac{3}{4}\right)^{3} \times\left(\frac{3}{4}\right)^{3}$ using the product rule.
$\left(\frac{3}{4}\right)^{3} \times\left(\frac{3}{4}\right)^{3} = \left(\frac{3}{4}\right)^{3+3}$

STEP 15

Calculate the exponent.
$\left(\frac{3}{4}\right)^{3+3} = \left(\frac{3}{4}\right)^{6}$

STEP 16

Find the value of $\left(\frac{3}{4}\right)^{6}$ .
$\left(\frac{3}{4}\right)^{6} = \left(\frac{3}{4}\right) \times \left(\frac{3}{4}\right) \times \left(\frac{3}{4}\right) \times \left(\frac{3}{4}\right) \times \left(\frac{3}{4}\right) \times \left(\frac{3}{4}\right)$

STEP 17

Calculate the value.
$\left(\frac{3}{4}\right)^{6} = \left(\frac{729}{4096}\right)$

STEP 18

Summarize the final values for each expression.
a) $4^{5} \times 4^{2} = 16384$ c) $(-2)^{2} \times(-2)^{4} \times(-2) = -128$ b) $2^{3} \times 2^{2} \times 2^{2} = 128$ d) $\left(\frac{3}{4}\right)^{3} \times\left(\frac{3}{4}\right)^{3} = \frac{729}{4096}$

Was this helpful?

Use the product rule to simplify and find the values of $4^{5} \times 4^{2}$ , $(-2)^{2} \times(-2)^{4} \times(-2)$ , $2^{3} \times 2^{2} \times 2^{2}$ , and $\left(\frac{3}{4}\right)^{3} \times\left(\frac{3}{4}\right)^{3}$ .

STEP 1

Assumptions
1. We will use the product rule of exponents which states that $a^m \times a^n = a^{m+n}$ .
2. We will simplify each expression using the product rule.
3. After simplifying, we will calculate the value of each expression.

STEP 2

Simplify the expression $4^{5} \times 4^{2}$ using the product rule.
$4^{5} \times 4^{2} = 4^{5+2}$

STEP 3

Calculate the exponent.
$4^{5+2} = 4^{7}$

STEP 4

Find the value of $4^{7}$ .
$4^{7} = 4 \times 4 \times 4 \times 4 \times 4 \times 4 \times 4$

STEP 5

Calculate the value.
$4^{7} = 16384$

STEP 6

Simplify the expression $(-2)^{2} \times(-2)^{4} \times(-2)$ using the product rule.
$(-2)^{2} \times(-2)^{4} \times(-2) = (-2)^{2+4+1}$

STEP 7

Calculate the exponent.
$(-2)^{2+4+1} = (-2)^{7}$

STEP 8

Find the value of $(-2)^{7}$ .
$(-2)^{7} = -2 \times -2 \times -2 \times -2 \times -2 \times -2 \times -2$

STEP 9

Calculate the value.
$(-2)^{7} = -128$

STEP 10

Simplify the expression $2^{3} \times 2^{2} \times 2^{2}$ using the product rule.
$2^{3} \times 2^{2} \times 2^{2} = 2^{3+2+2}$

STEP 11

Calculate the exponent.
$2^{3+2+2} = 2^{7}$

STEP 12

Find the value of $2^{7}$ .
$2^{7} = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2$

STEP 13

Calculate the value.
$2^{7} = 128$

STEP 14

Simplify the expression $\left(\frac{3}{4}\right)^{3} \times\left(\frac{3}{4}\right)^{3}$ using the product rule.
$\left(\frac{3}{4}\right)^{3} \times\left(\frac{3}{4}\right)^{3} = \left(\frac{3}{4}\right)^{3+3}$

STEP 15

Calculate the exponent.
$\left(\frac{3}{4}\right)^{3+3} = \left(\frac{3}{4}\right)^{6}$

STEP 16

Find the value of $\left(\frac{3}{4}\right)^{6}$ .
$\left(\frac{3}{4}\right)^{6} = \left(\frac{3}{4}\right) \times \left(\frac{3}{4}\right) \times \left(\frac{3}{4}\right) \times \left(\frac{3}{4}\right) \times \left(\frac{3}{4}\right) \times \left(\frac{3}{4}\right)$

STEP 17

Calculate the value.
$\left(\frac{3}{4}\right)^{6} = \left(\frac{729}{4096}\right)$

STEP 18

Summarize the final values for each expression.
a) $4^{5} \times 4^{2} = 16384$ c) $(-2)^{2} \times(-2)^{4} \times(-2) = -128$ b) $2^{3} \times 2^{2} \times 2^{2} = 128$ d) $\left(\frac{3}{4}\right)^{3} \times\left(\frac{3}{4}\right)^{3} = \frac{729}{4096}$

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

STEP 1

Assumptions1. We will use the product rule of exponents which states that am×an=am+na^m \times a^n = a^{m+n}am×an=am+n.2. We will simplify each expression using the product rule.3. After simplifying, we will calculate the value of each expression.

STEP 2

Simplify the expression 45×424^{5} \times 4^{2}45×42 using the product rule.45×42=45+24^{5} \times 4^{2} = 4^{5+2}45×42=45+2

STEP 3

Calculate the exponent.45+2=474^{5+2} = 4^{7}45+2=47

STEP 4

Find the value of 474^{7}47.47=4×4×4×4×4×4×44^{7} = 4 \times 4 \times 4 \times 4 \times 4 \times 4 \times 447=4×4×4×4×4×4×4

STEP 5

Calculate the value.47=163844^{7} = 1638447=16384

STEP 6

Simplify the expression (−2)2×(−2)4×(−2)(-2)^{2} \times(-2)^{4} \times(-2)(−2)2×(−2)4×(−2) using the product rule.(−2)2×(−2)4×(−2)=(−2)2+4+1(-2)^{2} \times(-2)^{4} \times(-2) = (-2)^{2+4+1}(−2)2×(−2)4×(−2)=(−2)2+4+1

STEP 7

Calculate the exponent.(−2)2+4+1=(−2)7(-2)^{2+4+1} = (-2)^{7}(−2)2+4+1=(−2)7

STEP 8

Find the value of (−2)7(-2)^{7}(−2)7.(−2)7=−2×−2×−2×−2×−2×−2×−2(-2)^{7} = -2 \times -2 \times -2 \times -2 \times -2 \times -2 \times -2(−2)7=−2×−2×−2×−2×−2×−2×−2

STEP 9

Calculate the value.(−2)7=−128(-2)^{7} = -128(−2)7=−128

STEP 10

Simplify the expression 23×22×222^{3} \times 2^{2} \times 2^{2}23×22×22 using the product rule.23×22×22=23+2+22^{3} \times 2^{2} \times 2^{2} = 2^{3+2+2}23×22×22=23+2+2

STEP 11

Calculate the exponent.23+2+2=272^{3+2+2} = 2^{7}23+2+2=27

STEP 12

Find the value of 272^{7}27.27=2×2×2×2×2×2×22^{7} = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 227=2×2×2×2×2×2×2

STEP 13

Calculate the value.27=1282^{7} = 12827=128

STEP 14

Simplify the expression (34)3×(34)3\left(\frac{3}{4}\right)^{3} \times\left(\frac{3}{4}\right)^{3}(43​)3×(43​)3 using the product rule.(34)3×(34)3=(34)3+3\left(\frac{3}{4}\right)^{3} \times\left(\frac{3}{4}\right)^{3} = \left(\frac{3}{4}\right)^{3+3}(43​)3×(43​)3=(43​)3+3

STEP 15

Calculate the exponent.(34)3+3=(34)6\left(\frac{3}{4}\right)^{3+3} = \left(\frac{3}{4}\right)^{6}(43​)3+3=(43​)6

STEP 16

STEP 17

Calculate the value.(34)6=(7294096)\left(\frac{3}{4}\right)^{6} = \left(\frac{729}{4096}\right)(43​)6=(4096729​)

STEP 18

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Assumptions
1. We will use the product rule of exponents which states that $a^m \times a^n = a^{m+n}$ .
2. We will simplify each expression using the product rule.
3. After simplifying, we will calculate the value of each expression.

Simplify the expression $4^{5} \times 4^{2}$ using the product rule.
$4^{5} \times 4^{2} = 4^{5+2}$

Calculate the exponent.
$4^{5+2} = 4^{7}$

Find the value of $4^{7}$ .
$4^{7} = 4 \times 4 \times 4 \times 4 \times 4 \times 4 \times 4$

Calculate the value.
$4^{7} = 16384$

Simplify the expression $(-2)^{2} \times(-2)^{4} \times(-2)$ using the product rule.
$(-2)^{2} \times(-2)^{4} \times(-2) = (-2)^{2+4+1}$

Calculate the exponent.
$(-2)^{2+4+1} = (-2)^{7}$

Find the value of $(-2)^{7}$ .
$(-2)^{7} = -2 \times -2 \times -2 \times -2 \times -2 \times -2 \times -2$

Calculate the value.
$(-2)^{7} = -128$

Simplify the expression $2^{3} \times 2^{2} \times 2^{2}$ using the product rule.
$2^{3} \times 2^{2} \times 2^{2} = 2^{3+2+2}$

Calculate the exponent.
$2^{3+2+2} = 2^{7}$

Find the value of $2^{7}$ .
$2^{7} = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2$

Calculate the value.
$2^{7} = 128$

Simplify the expression $\left(\frac{3}{4}\right)^{3} \times\left(\frac{3}{4}\right)^{3}$ using the product rule.
$\left(\frac{3}{4}\right)^{3} \times\left(\frac{3}{4}\right)^{3} = \left(\frac{3}{4}\right)^{3+3}$

Calculate the exponent.
$\left(\frac{3}{4}\right)^{3+3} = \left(\frac{3}{4}\right)^{6}$

Calculate the value.
$\left(\frac{3}{4}\right)^{6} = \left(\frac{729}{4096}\right)$