QuestionSimplify the following expressions:
94. $\frac{72 x^{\frac{3}{4}}}{9 x^{\frac{1}{3}}}$
96. $\left(x^{\frac{4}{5}}\right)^{5}$
98. $\left(125 x^{9} y^{6}\right)^{\frac{1}{3}}$
100. $\frac{\left(2 y^{\frac{1}{5}}\right)^{4}}{y^{\frac{3}{10}}}$

Studdy Solution

STEP 1

Assumptions1. We are given four different expressions to simplify. We will use the laws of exponents to simplify each expression3. The laws of exponents state that when multiplying like bases, you add the exponents, and when dividing like bases, you subtract the exponents

STEP 2

Let's start with the first expression $\frac{72 x^{\frac{}{4}}}{9 x^{\frac{1}{}}}$ We can simplify the constants and the variable terms separately.

STEP 3

implify the constants $\frac{72}{9} =8$

STEP 4

implify the variable term by subtracting the exponents $x^{\frac{3}{4} - \frac{1}{3}}$

STEP 5

Subtract the fractions $x^{\frac{9}{12} - \frac{4}{12}} = x^{\frac{5}{12}}$

STEP 6

Combine the constants and the variable term $8x^{\frac{5}{12}}$

STEP 7

Now, let's move to the second expression $\left(x^{\frac{4}{5}}\right)^{5}$ We simplify this by multiplying the exponents.

STEP 8

Multiply the exponents $x^{\frac{4}{5} \times5}$

STEP 9

implify the multiplication $x^{4}$

STEP 10

Next, we'll simplify the third expression $\left(125 x^{9} y^{6}\right)^{\frac{}{3}}$
We can apply the exponent to each part of the expression inside the parentheses.

STEP 11

Apply the exponent to the constant $125^{\frac{}{3}} =5$

STEP 12

Apply the exponent to the variable terms $x^{9 \times \frac{}{}} = x^{}$ $y^{6 \times \frac{}{}} = y^{2}$

STEP 13

Combine the constants and the variable terms $5x^{3}y^{2}$

STEP 14

Finally, we'll simplify the fourth expression $\frac{\left(2 y^{\frac{}{}}\right)^{4}}{y^{\frac{3}{10}}}$ We can simplify the numerator and the denominator separately.

STEP 15

implify the numerator by applying the exponent to the constant and the variable term $\left(2^{4}\right) \left(y^{\frac{}{5} \times4}\right) =y^{\frac{4}{5}}$

STEP 16

Now, we can simplify the expression by dividing the numerator by the denominator.

STEP 17

Divide the numerator by the denominator $\frac{16y^{\frac{4}{5}}}{y^{\frac{3}{10}}}$

STEP 18

implify the variable term by subtracting the exponents $y^{\frac{4}{5} - \frac{3}{10}}$

STEP 19

Subtract the fractions $y^{\frac{8}{10} - \frac{3}{10}} = y^{\frac{5}{10}} = y^{\frac{1}{}}$

STEP 20

Combine the constants and the variable term $16y^{\frac{}{}}$ So the simplified forms of the given expressions are94. $8x^{\frac{5}{12}}$
96. $x^{4}$
98. $5x^{3}y^{}$
100. $16y^{\frac{}{}}$

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Studdy Solution

STEP 1

Assumptions1. We are given four different expressions to simplify. We will use the laws of exponents to simplify each expression3. The laws of exponents state that when multiplying like bases, you add the exponents, and when dividing like bases, you subtract the exponents

STEP 2

Let's start with the first expression 72x49x1\frac{72 x^{\frac{}{4}}}{9 x^{\frac{1}{}}}9x1​72x4​​We can simplify the constants and the variable terms separately.

STEP 3

implify the constants729=8\frac{72}{9} =8972​=8

STEP 4

implify the variable term by subtracting the exponentsx34−13x^{\frac{3}{4} - \frac{1}{3}}x43​−31​

STEP 5

Subtract the fractionsx912−412=x512x^{\frac{9}{12} - \frac{4}{12}} = x^{\frac{5}{12}}x129​−124​=x125​

STEP 6

Combine the constants and the variable term8x5128x^{\frac{5}{12}}8x125​

STEP 7

Now, let's move to the second expression (x45)5\left(x^{\frac{4}{5}}\right)^{5}(x54​)5We simplify this by multiplying the exponents.

STEP 8

Multiply the exponentsx45×5x^{\frac{4}{5} \times5}x54​×5

STEP 9

implify the multiplicationx4x^{4}x4

STEP 10

Next, we'll simplify the third expression (125x9y6)3\left(125 x^{9} y^{6}\right)^{\frac{}{3}}(125x9y6)3​We can apply the exponent to each part of the expression inside the parentheses.

STEP 11

Apply the exponent to the constant1253=5125^{\frac{}{3}} =51253​=5

STEP 12

Apply the exponent to the variable termsx9×=xx^{9 \times \frac{}{}} = x^{}x9×​=xy6×=y2y^{6 \times \frac{}{}} = y^{2}y6×​=y2

STEP 13

Combine the constants and the variable terms5x3y25x^{3}y^{2}5x3y2

STEP 14

Finally, we'll simplify the fourth expression (2y)4y310\frac{\left(2 y^{\frac{}{}}\right)^{4}}{y^{\frac{3}{10}}}y103​(2y​)4​We can simplify the numerator and the denominator separately.

STEP 15

implify the numerator by applying the exponent to the constant and the variable term(24)(y5×4)=y45\left(2^{4}\right) \left(y^{\frac{}{5} \times4}\right) =y^{\frac{4}{5}}(24)(y5​×4)=y54​

STEP 16

Now, we can simplify the expression by dividing the numerator by the denominator.

STEP 17

Divide the numerator by the denominator16y45y310\frac{16y^{\frac{4}{5}}}{y^{\frac{3}{10}}}y103​16y54​​

STEP 18

implify the variable term by subtracting the exponentsy45−310y^{\frac{4}{5} - \frac{3}{10}}y54​−103​

STEP 19

Subtract the fractionsy810−310=y510=y1y^{\frac{8}{10} - \frac{3}{10}} = y^{\frac{5}{10}} = y^{\frac{1}{}}y108​−103​=y105​=y1​

STEP 20

Combine the constants and the variable term16y16y^{\frac{}{}}16y​So the simplified forms of the given expressions are94. 8x5128x^{\frac{5}{12}}8x125​96. x4x^{4}x498. 5x3y5x^{3}y^{}5x3y100. 16y16y^{\frac{}{}}16y​

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Let's start with the first expression $\frac{72 x^{\frac{}{4}}}{9 x^{\frac{1}{}}}$ We can simplify the constants and the variable terms separately.

implify the constants $\frac{72}{9} =8$

implify the variable term by subtracting the exponents $x^{\frac{3}{4} - \frac{1}{3}}$

Subtract the fractions $x^{\frac{9}{12} - \frac{4}{12}} = x^{\frac{5}{12}}$

Combine the constants and the variable term $8x^{\frac{5}{12}}$

Now, let's move to the second expression $\left(x^{\frac{4}{5}}\right)^{5}$ We simplify this by multiplying the exponents.

Multiply the exponents $x^{\frac{4}{5} \times5}$

implify the multiplication $x^{4}$

Next, we'll simplify the third expression $\left(125 x^{9} y^{6}\right)^{\frac{}{3}}$
We can apply the exponent to each part of the expression inside the parentheses.

Apply the exponent to the constant $125^{\frac{}{3}} =5$

Apply the exponent to the variable terms $x^{9 \times \frac{}{}} = x^{}$ $y^{6 \times \frac{}{}} = y^{2}$

Combine the constants and the variable terms $5x^{3}y^{2}$

Finally, we'll simplify the fourth expression $\frac{\left(2 y^{\frac{}{}}\right)^{4}}{y^{\frac{3}{10}}}$ We can simplify the numerator and the denominator separately.

implify the numerator by applying the exponent to the constant and the variable term $\left(2^{4}\right) \left(y^{\frac{}{5} \times4}\right) =y^{\frac{4}{5}}$

Divide the numerator by the denominator $\frac{16y^{\frac{4}{5}}}{y^{\frac{3}{10}}}$

implify the variable term by subtracting the exponents $y^{\frac{4}{5} - \frac{3}{10}}$

Subtract the fractions $y^{\frac{8}{10} - \frac{3}{10}} = y^{\frac{5}{10}} = y^{\frac{1}{}}$

Combine the constants and the variable term $16y^{\frac{}{}}$ So the simplified forms of the given expressions are94. $8x^{\frac{5}{12}}$
96. $x^{4}$
98. $5x^{3}y^{}$
100. $16y^{\frac{}{}}$