Basic Math Examples

$\sqrt{a} + \sqrt{b} = \sqrt{45}$

Step 1

Solve for $\sqrt{a}$ .

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Step 1.1

Simplify $\sqrt{45}$ .

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Step 1.1.1

Rewrite $45$ as $3^{2} \cdot 5$ .

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Step 1.1.1.1

Factor $9$ out of $45$ .

$\sqrt{a} + \sqrt{b} = \sqrt{9 (5)}$

Step 1.1.1.2

Rewrite $9$ as $3^{2}$ .

$\sqrt{a} + \sqrt{b} = \sqrt{3^{2} \cdot 5}$

Step 1.1.2

Pull terms out from under the radical.

$\sqrt{a} + \sqrt{b} = 3 \sqrt{5}$

Step 1.2

Subtract $\sqrt{b}$ from both sides of the equation.

$\sqrt{a} = 3 \sqrt{5} - \sqrt{b}$

Step 2

To remove the radical on the left side of the equation, square both sides of the equation.

${\sqrt{a}}^{2} = {(3 \sqrt{5} - \sqrt{b})}^{2}$

Step 3

Simplify each side of the equation.

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Step 3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{a}$ as $a^{\frac{1}{2}}$ .

${(a^{\frac{1}{2}})}^{2} = {(3 \sqrt{5} - \sqrt{b})}^{2}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Simplify ${(a^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1

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

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Step 3.2.1.1.1

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

$a^{\frac{1}{2} \cdot 2} = {(3 \sqrt{5} - \sqrt{b})}^{2}$

Step 3.2.1.1.2

Cancel the common factor of $2$ .

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Step 3.2.1.1.2.1

Cancel the common factor.

$a^{\frac{1}{2} \cdot 2} = {(3 \sqrt{5} - \sqrt{b})}^{2}$

Step 3.2.1.1.2.2

Rewrite the expression.

$a^{1} = {(3 \sqrt{5} - \sqrt{b})}^{2}$

Step 3.2.1.2

Simplify.

$a = {(3 \sqrt{5} - \sqrt{b})}^{2}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Simplify ${(3 \sqrt{5} - \sqrt{b})}^{2}$ .

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Step 3.3.1.1

Rewrite ${(3 \sqrt{5} - \sqrt{b})}^{2}$ as $(3 \sqrt{5} - \sqrt{b}) (3 \sqrt{5} - \sqrt{b})$ .

$a = (3 \sqrt{5} - \sqrt{b}) (3 \sqrt{5} - \sqrt{b})$

Step 3.3.1.2

Expand $(3 \sqrt{5} - \sqrt{b}) (3 \sqrt{5} - \sqrt{b})$ using the FOIL Method.

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Step 3.3.1.2.1

Apply the distributive property.

$a = 3 \sqrt{5} (3 \sqrt{5} - \sqrt{b}) - \sqrt{b} (3 \sqrt{5} - \sqrt{b})$

Step 3.3.1.2.2

Apply the distributive property.

$a = 3 \sqrt{5} (3 \sqrt{5}) + 3 \sqrt{5} (- \sqrt{b}) - \sqrt{b} (3 \sqrt{5} - \sqrt{b})$

Step 3.3.1.2.3

Apply the distributive property.

$a = 3 \sqrt{5} (3 \sqrt{5}) + 3 \sqrt{5} (- \sqrt{b}) - \sqrt{b} (3 \sqrt{5}) - \sqrt{b} (- \sqrt{b})$

Step 3.3.1.3

Simplify and combine like terms.

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Step 3.3.1.3.1

Simplify each term.

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Step 3.3.1.3.1.1

Multiply $3 \sqrt{5} (3 \sqrt{5})$ .

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Step 3.3.1.3.1.1.1

Multiply $3$ by $3$ .

$a = 9 \sqrt{5} \sqrt{5} + 3 \sqrt{5} (- \sqrt{b}) - \sqrt{b} (3 \sqrt{5}) - \sqrt{b} (- \sqrt{b})$

Step 3.3.1.3.1.1.2

Raise $\sqrt{5}$ to the power of $1$ .

$a = 9 ({\sqrt{5}}^{1} \sqrt{5}) + 3 \sqrt{5} (- \sqrt{b}) - \sqrt{b} (3 \sqrt{5}) - \sqrt{b} (- \sqrt{b})$

Step 3.3.1.3.1.1.3

Raise $\sqrt{5}$ to the power of $1$ .

$a = 9 ({\sqrt{5}}^{1} {\sqrt{5}}^{1}) + 3 \sqrt{5} (- \sqrt{b}) - \sqrt{b} (3 \sqrt{5}) - \sqrt{b} (- \sqrt{b})$

Step 3.3.1.3.1.1.4

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

$a = 9 {\sqrt{5}}^{1 + 1} + 3 \sqrt{5} (- \sqrt{b}) - \sqrt{b} (3 \sqrt{5}) - \sqrt{b} (- \sqrt{b})$

Step 3.3.1.3.1.1.5

Add $1$ and $1$ .

$a = 9 {\sqrt{5}}^{2} + 3 \sqrt{5} (- \sqrt{b}) - \sqrt{b} (3 \sqrt{5}) - \sqrt{b} (- \sqrt{b})$

Step 3.3.1.3.1.2

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

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Step 3.3.1.3.1.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$a = 9 {(5^{\frac{1}{2}})}^{2} + 3 \sqrt{5} (- \sqrt{b}) - \sqrt{b} (3 \sqrt{5}) - \sqrt{b} (- \sqrt{b})$

Step 3.3.1.3.1.2.2

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

$a = 9 \cdot 5^{\frac{1}{2} \cdot 2} + 3 \sqrt{5} (- \sqrt{b}) - \sqrt{b} (3 \sqrt{5}) - \sqrt{b} (- \sqrt{b})$

Step 3.3.1.3.1.2.3

Combine $\frac{1}{2}$ and $2$ .

$a = 9 \cdot 5^{\frac{2}{2}} + 3 \sqrt{5} (- \sqrt{b}) - \sqrt{b} (3 \sqrt{5}) - \sqrt{b} (- \sqrt{b})$

Step 3.3.1.3.1.2.4

Cancel the common factor of $2$ .

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Step 3.3.1.3.1.2.4.1

Cancel the common factor.

$a = 9 \cdot 5^{\frac{2}{2}} + 3 \sqrt{5} (- \sqrt{b}) - \sqrt{b} (3 \sqrt{5}) - \sqrt{b} (- \sqrt{b})$

Step 3.3.1.3.1.2.4.2

Rewrite the expression.

$a = 9 \cdot 5^{1} + 3 \sqrt{5} (- \sqrt{b}) - \sqrt{b} (3 \sqrt{5}) - \sqrt{b} (- \sqrt{b})$

Step 3.3.1.3.1.2.5

Evaluate the exponent.

$a = 9 \cdot 5 + 3 \sqrt{5} (- \sqrt{b}) - \sqrt{b} (3 \sqrt{5}) - \sqrt{b} (- \sqrt{b})$

Step 3.3.1.3.1.3

Multiply $9$ by $5$ .

$a = 45 + 3 \sqrt{5} (- \sqrt{b}) - \sqrt{b} (3 \sqrt{5}) - \sqrt{b} (- \sqrt{b})$

Step 3.3.1.3.1.4

Multiply $3 \sqrt{5} (- \sqrt{b})$ .

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Step 3.3.1.3.1.4.1

Multiply $- 1$ by $3$ .

$a = 45 - 3 \sqrt{5} \sqrt{b} - \sqrt{b} (3 \sqrt{5}) - \sqrt{b} (- \sqrt{b})$

Step 3.3.1.3.1.4.2

Combine using the product rule for radicals.

$a = 45 - 3 \sqrt{b \cdot 5} - \sqrt{b} (3 \sqrt{5}) - \sqrt{b} (- \sqrt{b})$

Step 3.3.1.3.1.5

Multiply $- \sqrt{b} (3 \sqrt{5})$ .

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Step 3.3.1.3.1.5.1

Multiply $3$ by $- 1$ .

$a = 45 - 3 \sqrt{b \cdot 5} - 3 \sqrt{b} \sqrt{5} - \sqrt{b} (- \sqrt{b})$

Step 3.3.1.3.1.5.2

Combine using the product rule for radicals.

$a = 45 - 3 \sqrt{b \cdot 5} - 3 \sqrt{5 b} - \sqrt{b} (- \sqrt{b})$

Step 3.3.1.3.1.6

Multiply $- \sqrt{b} (- \sqrt{b})$ .

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Step 3.3.1.3.1.6.1

Multiply $- 1$ by $- 1$ .

$a = 45 - 3 \sqrt{b \cdot 5} - 3 \sqrt{5 b} + 1 \sqrt{b} \sqrt{b}$

Step 3.3.1.3.1.6.2

Multiply $\sqrt{b}$ by $1$ .

$a = 45 - 3 \sqrt{b \cdot 5} - 3 \sqrt{5 b} + \sqrt{b} \sqrt{b}$

Step 3.3.1.3.1.6.3

Raise $\sqrt{b}$ to the power of $1$ .

$a = 45 - 3 \sqrt{b \cdot 5} - 3 \sqrt{5 b} + {\sqrt{b}}^{1} \sqrt{b}$

Step 3.3.1.3.1.6.4

Raise $\sqrt{b}$ to the power of $1$ .

$a = 45 - 3 \sqrt{b \cdot 5} - 3 \sqrt{5 b} + {\sqrt{b}}^{1} {\sqrt{b}}^{1}$

Step 3.3.1.3.1.6.5

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

$a = 45 - 3 \sqrt{b \cdot 5} - 3 \sqrt{5 b} + {\sqrt{b}}^{1 + 1}$

Step 3.3.1.3.1.6.6

Add $1$ and $1$ .

$a = 45 - 3 \sqrt{b \cdot 5} - 3 \sqrt{5 b} + {\sqrt{b}}^{2}$

Step 3.3.1.3.1.7

Rewrite ${\sqrt{b}}^{2}$ as $b$ .

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Step 3.3.1.3.1.7.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{b}$ as $b^{\frac{1}{2}}$ .

$a = 45 - 3 \sqrt{b \cdot 5} - 3 \sqrt{5 b} + {(b^{\frac{1}{2}})}^{2}$

Step 3.3.1.3.1.7.2

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

$a = 45 - 3 \sqrt{b \cdot 5} - 3 \sqrt{5 b} + b^{\frac{1}{2} \cdot 2}$

Step 3.3.1.3.1.7.3

Combine $\frac{1}{2}$ and $2$ .

$a = 45 - 3 \sqrt{b \cdot 5} - 3 \sqrt{5 b} + b^{\frac{2}{2}}$

Step 3.3.1.3.1.7.4

Cancel the common factor of $2$ .

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Step 3.3.1.3.1.7.4.1

Cancel the common factor.

$a = 45 - 3 \sqrt{b \cdot 5} - 3 \sqrt{5 b} + b^{\frac{2}{2}}$

Step 3.3.1.3.1.7.4.2

Rewrite the expression.

$a = 45 - 3 \sqrt{b \cdot 5} - 3 \sqrt{5 b} + b^{1}$

Step 3.3.1.3.1.7.5

Simplify.

$a = 45 - 3 \sqrt{b \cdot 5} - 3 \sqrt{5 b} + b$

Step 3.3.1.3.2

Subtract $3 \sqrt{5 b}$ from $- 3 \sqrt{b \cdot 5}$ .

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Step 3.3.1.3.2.1

Reorder $b$ and $5$ .

$a = 45 - 3 \sqrt{5 \cdot b} - 3 \sqrt{5 b} + b$

Step 3.3.1.3.2.2

Subtract $3 \sqrt{5 b}$ from $- 3 \sqrt{5 \cdot b}$ .

$a = 45 - 6 \sqrt{5 \cdot b} + b$