QuestionSimplify the expression: $64 a^{5} b^{3} \cdot\left(\frac{1}{2} a b^{3}\right)^{4}$ .

Studdy Solution

STEP 1

Assumptions1. We are given the expression $64 a^{5} b^{3} \cdot\left(\frac{1}{} a b^{3}\right)^{4}$ . We are assuming that all variables are real numbers3. We are assuming that the exponent rules apply, where $(a^m)^n = a^{mn}$ and $a^m \cdot a^n = a^{m+n}$

STEP 2

First, we need to simplify the expression inside the brackets and raise it to the power of4. We can do this by applying the exponent rule $(a^m)^n = a^{mn}$ .
$\left(\frac{1}{2} a b^{}\right)^{4} = \left(\frac{1}{2}\right)^{4} \cdot a^{4} \cdot (b^{})^{4}$

STEP 3

Now, simplify the expression further by calculating the power of each term.
$\left(\frac{1}{2}\right)^{} \cdot a^{} \cdot (b^{3})^{} = \frac{1}{16} \cdot a^{} \cdot b^{12}$

STEP 4

Now, replace the simplified expression back into the original expression.
$64 a^{} b^{3} \cdot\left(\frac{1}{2} a b^{3}\right)^{4} =64 a^{} b^{3} \cdot \frac{1}{16} \cdot a^{4} \cdot b^{12}$

STEP 5

Next, we simplify the expression by multiplying the constants and adding the powers of the same bases.
$64 a^{5} b^{3} \cdot \frac{1}{16} \cdot a^{4} \cdot b^{12} =4 a^{5+4} b^{3+12}$

STEP 6

Finally, simplify the expression by performing the addition in the exponents.
$4 a^{5+4} b^{3+12} =4 a^{9} b^{15}$ So, $64 a^{5} b^{3} \cdot\left(\frac{1}{2} a b^{3}\right)^{4}$ simplifies to $4 a^{9} b^{15}$ .

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QuestionSimplify the expression: $64 a^{5} b^{3} \cdot\left(\frac{1}{2} a b^{3}\right)^{4}$ .

Studdy Solution

STEP 1

Assumptions1. We are given the expression $64 a^{5} b^{3} \cdot\left(\frac{1}{} a b^{3}\right)^{4}$ . We are assuming that all variables are real numbers3. We are assuming that the exponent rules apply, where $(a^m)^n = a^{mn}$ and $a^m \cdot a^n = a^{m+n}$

STEP 2

First, we need to simplify the expression inside the brackets and raise it to the power of4. We can do this by applying the exponent rule $(a^m)^n = a^{mn}$ .
$\left(\frac{1}{2} a b^{}\right)^{4} = \left(\frac{1}{2}\right)^{4} \cdot a^{4} \cdot (b^{})^{4}$

STEP 3

Now, simplify the expression further by calculating the power of each term.
$\left(\frac{1}{2}\right)^{} \cdot a^{} \cdot (b^{3})^{} = \frac{1}{16} \cdot a^{} \cdot b^{12}$

STEP 4

Now, replace the simplified expression back into the original expression.
$64 a^{} b^{3} \cdot\left(\frac{1}{2} a b^{3}\right)^{4} =64 a^{} b^{3} \cdot \frac{1}{16} \cdot a^{4} \cdot b^{12}$

STEP 5

Next, we simplify the expression by multiplying the constants and adding the powers of the same bases.
$64 a^{5} b^{3} \cdot \frac{1}{16} \cdot a^{4} \cdot b^{12} =4 a^{5+4} b^{3+12}$

STEP 6

Finally, simplify the expression by performing the addition in the exponents.
$4 a^{5+4} b^{3+12} =4 a^{9} b^{15}$ So, $64 a^{5} b^{3} \cdot\left(\frac{1}{2} a b^{3}\right)^{4}$ simplifies to $4 a^{9} b^{15}$ .

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QuestionSimplify the expression: 64a5b3⋅(12ab3)464 a^{5} b^{3} \cdot\left(\frac{1}{2} a b^{3}\right)^{4}64a5b3⋅(21​ab3)4.

Studdy Solution

STEP 1

STEP 2

STEP 3

Now, simplify the expression further by calculating the power of each term.(12)⋅a⋅(b3)=116⋅a⋅b12\left(\frac{1}{2}\right)^{} \cdot a^{} \cdot (b^{3})^{} = \frac{1}{16} \cdot a^{} \cdot b^{12}(21​)⋅a⋅(b3)=161​⋅a⋅b12

STEP 4

Now, replace the simplified expression back into the original expression.64ab3⋅(12ab3)4=64ab3⋅116⋅a4⋅b1264 a^{} b^{3} \cdot\left(\frac{1}{2} a b^{3}\right)^{4} =64 a^{} b^{3} \cdot \frac{1}{16} \cdot a^{4} \cdot b^{12}64ab3⋅(21​ab3)4=64ab3⋅161​⋅a4⋅b12

STEP 5

Next, we simplify the expression by multiplying the constants and adding the powers of the same bases.64a5b3⋅116⋅a4⋅b12=4a5+4b3+1264 a^{5} b^{3} \cdot \frac{1}{16} \cdot a^{4} \cdot b^{12} =4 a^{5+4} b^{3+12}64a5b3⋅161​⋅a4⋅b12=4a5+4b3+12

STEP 6

Finally, simplify the expression by performing the addition in the exponents.4a5+4b3+12=4a9b154 a^{5+4} b^{3+12} =4 a^{9} b^{15}4a5+4b3+12=4a9b15So, 64a5b3⋅(12ab3)464 a^{5} b^{3} \cdot\left(\frac{1}{2} a b^{3}\right)^{4}64a5b3⋅(21​ab3)4 simplifies to 4a9b154 a^{9} b^{15}4a9b15.

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QuestionSimplify the expression: $64 a^{5} b^{3} \cdot\left(\frac{1}{2} a b^{3}\right)^{4}$ .

Now, simplify the expression further by calculating the power of each term.
$\left(\frac{1}{2}\right)^{} \cdot a^{} \cdot (b^{3})^{} = \frac{1}{16} \cdot a^{} \cdot b^{12}$

Now, replace the simplified expression back into the original expression.
$64 a^{} b^{3} \cdot\left(\frac{1}{2} a b^{3}\right)^{4} =64 a^{} b^{3} \cdot \frac{1}{16} \cdot a^{4} \cdot b^{12}$

Next, we simplify the expression by multiplying the constants and adding the powers of the same bases.
$64 a^{5} b^{3} \cdot \frac{1}{16} \cdot a^{4} \cdot b^{12} =4 a^{5+4} b^{3+12}$

Finally, simplify the expression by performing the addition in the exponents.
$4 a^{5+4} b^{3+12} =4 a^{9} b^{15}$ So, $64 a^{5} b^{3} \cdot\left(\frac{1}{2} a b^{3}\right)^{4}$ simplifies to $4 a^{9} b^{15}$ .