Algebra Examples

${(2 \frac{1}{3} x^{2 m - \frac{1}{2}})}^{2} \cdot {(\frac{3}{7} x^{\frac{1}{2}})}^{2}$

Step 1

Convert $2 \frac{1}{3}$ to an improper fraction.

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Step 1.1

A mixed number is an addition of its whole and fractional parts.

${((2 + \frac{1}{3}) x^{2 m - \frac{1}{2}})}^{2} \cdot {(\frac{3}{7} x^{\frac{1}{2}})}^{2}$

Step 1.2

Add $2$ and $\frac{1}{3}$ .

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Step 1.2.1

To write $2$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

${((2 \cdot \frac{3}{3} + \frac{1}{3}) x^{2 m - \frac{1}{2}})}^{2} \cdot {(\frac{3}{7} x^{\frac{1}{2}})}^{2}$

Step 1.2.2

Combine $2$ and $\frac{3}{3}$ .

${((\frac{2 \cdot 3}{3} + \frac{1}{3}) x^{2 m - \frac{1}{2}})}^{2} \cdot {(\frac{3}{7} x^{\frac{1}{2}})}^{2}$

Step 1.2.3

Combine the numerators over the common denominator.

${(\frac{2 \cdot 3 + 1}{3} x^{2 m - \frac{1}{2}})}^{2} \cdot {(\frac{3}{7} x^{\frac{1}{2}})}^{2}$

Step 1.2.4

Simplify the numerator.

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Step 1.2.4.1

Multiply $2$ by $3$ .

${(\frac{6 + 1}{3} x^{2 m - \frac{1}{2}})}^{2} \cdot {(\frac{3}{7} x^{\frac{1}{2}})}^{2}$

Step 1.2.4.2

Add $6$ and $1$ .

${(\frac{7}{3} x^{2 m - \frac{1}{2}})}^{2} \cdot {(\frac{3}{7} x^{\frac{1}{2}})}^{2}$

Step 2

Combine $\frac{7}{3}$ and $x^{2 m - \frac{1}{2}}$ .

${(\frac{7 x^{2 m - \frac{1}{2}}}{3})}^{2} \cdot {(\frac{3}{7} x^{\frac{1}{2}})}^{2}$

Step 3

Simplify the numerator.

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Step 3.1

To write $2 m$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

${(\frac{7 x^{2 m \cdot \frac{2}{2} - \frac{1}{2}}}{3})}^{2} \cdot {(\frac{3}{7} x^{\frac{1}{2}})}^{2}$

Step 3.2

Combine $2 m$ and $\frac{2}{2}$ .

${(\frac{7 x^{\frac{2 m \cdot 2}{2} - \frac{1}{2}}}{3})}^{2} \cdot {(\frac{3}{7} x^{\frac{1}{2}})}^{2}$

Step 3.3

Combine the numerators over the common denominator.

${(\frac{7 x^{\frac{2 m \cdot 2 - 1}{2}}}{3})}^{2} \cdot {(\frac{3}{7} x^{\frac{1}{2}})}^{2}$

Step 3.4

Multiply $2$ by $2$ .

${(\frac{7 x^{\frac{4 m - 1}{2}}}{3})}^{2} \cdot {(\frac{3}{7} x^{\frac{1}{2}})}^{2}$

Step 4

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 4.1

Apply the product rule to $\frac{7 x^{\frac{4 m - 1}{2}}}{3}$ .

$\frac{{(7 x^{\frac{4 m - 1}{2}})}^{2}}{3^{2}} \cdot {(\frac{3}{7} x^{\frac{1}{2}})}^{2}$

Step 4.2

Apply the product rule to $7 x^{\frac{4 m - 1}{2}}$ .

$\frac{7^{2} {(x^{\frac{4 m - 1}{2}})}^{2}}{3^{2}} \cdot {(\frac{3}{7} x^{\frac{1}{2}})}^{2}$

Step 5

Simplify the numerator.

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Step 5.1

Raise $7$ to the power of $2$ .

$\frac{49 {(x^{\frac{4 m - 1}{2}})}^{2}}{3^{2}} \cdot {(\frac{3}{7} x^{\frac{1}{2}})}^{2}$

Step 5.2

Multiply the exponents in ${(x^{\frac{4 m - 1}{2}})}^{2}$ .

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Step 5.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{49 x^{\frac{4 m - 1}{2} \cdot 2}}{3^{2}} \cdot {(\frac{3}{7} x^{\frac{1}{2}})}^{2}$

Step 5.2.2

Cancel the common factor of $2$ .

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Step 5.2.2.1

Cancel the common factor.

$\frac{49 x^{\frac{4 m - 1}{2} \cdot 2}}{3^{2}} \cdot {(\frac{3}{7} x^{\frac{1}{2}})}^{2}$

Step 5.2.2.2

Rewrite the expression.

$\frac{49 x^{4 m - 1}}{3^{2}} \cdot {(\frac{3}{7} x^{\frac{1}{2}})}^{2}$

Step 6

Combine fractions.

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Step 6.1

Raise $3$ to the power of $2$ .

$\frac{49 x^{4 m - 1}}{9} \cdot {(\frac{3}{7} x^{\frac{1}{2}})}^{2}$

Step 6.2

Combine $\frac{3}{7}$ and $x^{\frac{1}{2}}$ .

$\frac{49 x^{4 m - 1}}{9} \cdot {(\frac{3 x^{\frac{1}{2}}}{7})}^{2}$

Step 7

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 7.1

Apply the product rule to $\frac{3 x^{\frac{1}{2}}}{7}$ .

$\frac{49 x^{4 m - 1}}{9} \cdot \frac{{(3 x^{\frac{1}{2}})}^{2}}{7^{2}}$

Step 7.2

Apply the product rule to $3 x^{\frac{1}{2}}$ .

$\frac{49 x^{4 m - 1}}{9} \cdot \frac{3^{2} {(x^{\frac{1}{2}})}^{2}}{7^{2}}$

Step 8

Combine.

$\frac{49 x^{4 m - 1} (3^{2} {(x^{\frac{1}{2}})}^{2})}{9 \cdot 7^{2}}$

Step 9

Simplify the numerator.

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Step 9.1

Raise $3$ to the power of $2$ .

$\frac{49 x^{4 m - 1} \cdot 9 {(x^{\frac{1}{2}})}^{2}}{9 \cdot 7^{2}}$

Step 9.2

Multiply the exponents in ${(x^{\frac{1}{2}})}^{2}$ .

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Step 9.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{49 x^{4 m - 1} \cdot 9 x^{\frac{1}{2} \cdot 2}}{9 \cdot 7^{2}}$

Step 9.2.2

Cancel the common factor of $2$ .

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Step 9.2.2.1

Cancel the common factor.

$\frac{49 x^{4 m - 1} \cdot 9 x^{\frac{1}{2} \cdot 2}}{9 \cdot 7^{2}}$

Step 9.2.2.2

Rewrite the expression.

$\frac{49 x^{4 m - 1} \cdot 9 x^{1}}{9 \cdot 7^{2}}$

Step 9.3

Simplify.

$\frac{49 x^{4 m - 1} \cdot 9 x}{9 \cdot 7^{2}}$

Step 9.4

Combine exponents.

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Step 9.4.1

Multiply $9$ by $49$ .

$\frac{441 x^{4 m - 1} x}{9 \cdot 7^{2}}$

Step 9.4.2

Raise $x$ to the power of $1$ .

$\frac{441 (x^{1} x^{4 m - 1})}{9 \cdot 7^{2}}$

Step 9.4.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{441 x^{1 + 4 m - 1}}{9 \cdot 7^{2}}$

Step 9.4.4

Subtract $1$ from $1$ .

$\frac{441 x^{4 m + 0}}{9 \cdot 7^{2}}$

Step 9.4.5

Add $4 m$ and $0$ .

$\frac{441 x^{4 m}}{9 \cdot 7^{2}}$

Step 10

Reduce the expression by cancelling the common factors.

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Step 10.1

Raise $7$ to the power of $2$ .

$\frac{441 x^{4 m}}{9 \cdot 49}$

Step 10.2

Multiply $9$ by $49$ .

$\frac{441 x^{4 m}}{441}$

Step 10.3

Cancel the common factor of $441$ .

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Step 10.3.1

Cancel the common factor.

$\frac{441 x^{4 m}}{441}$

Step 10.3.2

Divide $x^{4 m}$ by $1$ .

$x^{4 m}$