Linear Algebra Examples

$2 P \sqrt{\frac{B R A G}{X y}} = M$

Step 1

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

${(2 P \sqrt{\frac{B R A G}{X y}})}^{2} = M^{2}$

Step 2

Simplify each side of the equation.

Tap for more steps...

Step 2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{\frac{B R A G}{X y}}$ as ${(\frac{B R A G}{X y})}^{\frac{1}{2}}$ .

${(2 P {(\frac{B R A G}{X y})}^{\frac{1}{2}})}^{2} = M^{2}$

Step 2.2

Simplify the left side.

Tap for more steps...

Step 2.2.1

Simplify ${(2 P {(\frac{B R A G}{X y})}^{\frac{1}{2}})}^{2}$ .

Tap for more steps...

Step 2.2.1.1

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

Tap for more steps...

Step 2.2.1.1.1

Apply the product rule to $\frac{B R A G}{X y}$ .

${(2 P \frac{{(B R A G)}^{\frac{1}{2}}}{{(X y)}^{\frac{1}{2}}})}^{2} = M^{2}$

Step 2.2.1.1.2

Apply the product rule to $B R A G$ .

${(2 P \frac{{(B R A)}^{\frac{1}{2}} G^{\frac{1}{2}}}{{(X y)}^{\frac{1}{2}}})}^{2} = M^{2}$

Step 2.2.1.1.3

Apply the product rule to $B R A$ .

${(2 P \frac{{(B R)}^{\frac{1}{2}} A^{\frac{1}{2}} G^{\frac{1}{2}}}{{(X y)}^{\frac{1}{2}}})}^{2} = M^{2}$

Step 2.2.1.1.4

Apply the product rule to $B R$ .

${(2 P \frac{B^{\frac{1}{2}} R^{\frac{1}{2}} A^{\frac{1}{2}} G^{\frac{1}{2}}}{{(X y)}^{\frac{1}{2}}})}^{2} = M^{2}$

Step 2.2.1.1.5

Apply the product rule to $X y$ .

${(2 P \frac{B^{\frac{1}{2}} R^{\frac{1}{2}} A^{\frac{1}{2}} G^{\frac{1}{2}}}{X^{\frac{1}{2}} y^{\frac{1}{2}}})}^{2} = M^{2}$

Step 2.2.1.2

Multiply $2 P \frac{B^{\frac{1}{2}} R^{\frac{1}{2}} A^{\frac{1}{2}} G^{\frac{1}{2}}}{X^{\frac{1}{2}} y^{\frac{1}{2}}}$ .

Tap for more steps...

Step 2.2.1.2.1

Combine $2$ and $\frac{B^{\frac{1}{2}} R^{\frac{1}{2}} A^{\frac{1}{2}} G^{\frac{1}{2}}}{X^{\frac{1}{2}} y^{\frac{1}{2}}}$ .

${(P \frac{2 (B^{\frac{1}{2}} R^{\frac{1}{2}} A^{\frac{1}{2}} G^{\frac{1}{2}})}{X^{\frac{1}{2}} y^{\frac{1}{2}}})}^{2} = M^{2}$

Step 2.2.1.2.2

Combine $P$ and $\frac{2 (B^{\frac{1}{2}} R^{\frac{1}{2}} A^{\frac{1}{2}} G^{\frac{1}{2}})}{X^{\frac{1}{2}} y^{\frac{1}{2}}}$ .

${(\frac{P (2 (B^{\frac{1}{2}} R^{\frac{1}{2}} A^{\frac{1}{2}} G^{\frac{1}{2}}))}{X^{\frac{1}{2}} y^{\frac{1}{2}}})}^{2} = M^{2}$

Step 2.2.1.3

Move $2$ to the left of $P$ .

${(\frac{2 \cdot P B^{\frac{1}{2}} R^{\frac{1}{2}} A^{\frac{1}{2}} G^{\frac{1}{2}}}{X^{\frac{1}{2}} y^{\frac{1}{2}}})}^{2} = M^{2}$

Step 2.2.1.4

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

Tap for more steps...

Step 2.2.1.4.1

Apply the product rule to $\frac{2 P B^{\frac{1}{2}} R^{\frac{1}{2}} A^{\frac{1}{2}} G^{\frac{1}{2}}}{X^{\frac{1}{2}} y^{\frac{1}{2}}}$ .

$\frac{{(2 P B^{\frac{1}{2}} R^{\frac{1}{2}} A^{\frac{1}{2}} G^{\frac{1}{2}})}^{2}}{{(X^{\frac{1}{2}} y^{\frac{1}{2}})}^{2}} = M^{2}$

Step 2.2.1.4.2

Apply the product rule to $2 P B^{\frac{1}{2}} R^{\frac{1}{2}} A^{\frac{1}{2}} G^{\frac{1}{2}}$ .

$\frac{{(2 P B^{\frac{1}{2}} R^{\frac{1}{2}} A^{\frac{1}{2}})}^{2} {(G^{\frac{1}{2}})}^{2}}{{(X^{\frac{1}{2}} y^{\frac{1}{2}})}^{2}} = M^{2}$

Step 2.2.1.4.3

Apply the product rule to $2 P B^{\frac{1}{2}} R^{\frac{1}{2}} A^{\frac{1}{2}}$ .

$\frac{{(2 P B^{\frac{1}{2}} R^{\frac{1}{2}})}^{2} {(A^{\frac{1}{2}})}^{2} {(G^{\frac{1}{2}})}^{2}}{{(X^{\frac{1}{2}} y^{\frac{1}{2}})}^{2}} = M^{2}$

Step 2.2.1.4.4

Apply the product rule to $2 P B^{\frac{1}{2}} R^{\frac{1}{2}}$ .

$\frac{{(2 P B^{\frac{1}{2}})}^{2} {(R^{\frac{1}{2}})}^{2} {(A^{\frac{1}{2}})}^{2} {(G^{\frac{1}{2}})}^{2}}{{(X^{\frac{1}{2}} y^{\frac{1}{2}})}^{2}} = M^{2}$

Step 2.2.1.4.5

Apply the product rule to $2 P B^{\frac{1}{2}}$ .

$\frac{{(2 P)}^{2} {(B^{\frac{1}{2}})}^{2} {(R^{\frac{1}{2}})}^{2} {(A^{\frac{1}{2}})}^{2} {(G^{\frac{1}{2}})}^{2}}{{(X^{\frac{1}{2}} y^{\frac{1}{2}})}^{2}} = M^{2}$

Step 2.2.1.4.6

Apply the product rule to $2 P$ .

$\frac{2^{2} P^{2} {(B^{\frac{1}{2}})}^{2} {(R^{\frac{1}{2}})}^{2} {(A^{\frac{1}{2}})}^{2} {(G^{\frac{1}{2}})}^{2}}{{(X^{\frac{1}{2}} y^{\frac{1}{2}})}^{2}} = M^{2}$

Step 2.2.1.4.7

Apply the product rule to $X^{\frac{1}{2}} y^{\frac{1}{2}}$ .

$\frac{2^{2} P^{2} {(B^{\frac{1}{2}})}^{2} {(R^{\frac{1}{2}})}^{2} {(A^{\frac{1}{2}})}^{2} {(G^{\frac{1}{2}})}^{2}}{{(X^{\frac{1}{2}})}^{2} {(y^{\frac{1}{2}})}^{2}} = M^{2}$

Step 2.2.1.5

Simplify the numerator.

Tap for more steps...

Step 2.2.1.5.1

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

$\frac{4 P^{2} {(B^{\frac{1}{2}})}^{2} {(R^{\frac{1}{2}})}^{2} {(A^{\frac{1}{2}})}^{2} {(G^{\frac{1}{2}})}^{2}}{{(X^{\frac{1}{2}})}^{2} {(y^{\frac{1}{2}})}^{2}} = M^{2}$

Step 2.2.1.5.2

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

Tap for more steps...

Step 2.2.1.5.2.1

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

$\frac{4 P^{2} B^{\frac{1}{2} \cdot 2} {(R^{\frac{1}{2}})}^{2} {(A^{\frac{1}{2}})}^{2} {(G^{\frac{1}{2}})}^{2}}{{(X^{\frac{1}{2}})}^{2} {(y^{\frac{1}{2}})}^{2}} = M^{2}$

Step 2.2.1.5.2.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 2.2.1.5.2.2.1

Cancel the common factor.

$\frac{4 P^{2} B^{\frac{1}{2} \cdot 2} {(R^{\frac{1}{2}})}^{2} {(A^{\frac{1}{2}})}^{2} {(G^{\frac{1}{2}})}^{2}}{{(X^{\frac{1}{2}})}^{2} {(y^{\frac{1}{2}})}^{2}} = M^{2}$

Step 2.2.1.5.2.2.2

Rewrite the expression.

$\frac{4 P^{2} B^{1} {(R^{\frac{1}{2}})}^{2} {(A^{\frac{1}{2}})}^{2} {(G^{\frac{1}{2}})}^{2}}{{(X^{\frac{1}{2}})}^{2} {(y^{\frac{1}{2}})}^{2}} = M^{2}$

Step 2.2.1.5.3

Simplify.

$\frac{4 P^{2} B {(R^{\frac{1}{2}})}^{2} {(A^{\frac{1}{2}})}^{2} {(G^{\frac{1}{2}})}^{2}}{{(X^{\frac{1}{2}})}^{2} {(y^{\frac{1}{2}})}^{2}} = M^{2}$

Step 2.2.1.5.4

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

Tap for more steps...

Step 2.2.1.5.4.1

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

$\frac{4 P^{2} B R^{\frac{1}{2} \cdot 2} {(A^{\frac{1}{2}})}^{2} {(G^{\frac{1}{2}})}^{2}}{{(X^{\frac{1}{2}})}^{2} {(y^{\frac{1}{2}})}^{2}} = M^{2}$

Step 2.2.1.5.4.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 2.2.1.5.4.2.1

Cancel the common factor.

$\frac{4 P^{2} B R^{\frac{1}{2} \cdot 2} {(A^{\frac{1}{2}})}^{2} {(G^{\frac{1}{2}})}^{2}}{{(X^{\frac{1}{2}})}^{2} {(y^{\frac{1}{2}})}^{2}} = M^{2}$

Step 2.2.1.5.4.2.2

Rewrite the expression.

$\frac{4 P^{2} B R^{1} {(A^{\frac{1}{2}})}^{2} {(G^{\frac{1}{2}})}^{2}}{{(X^{\frac{1}{2}})}^{2} {(y^{\frac{1}{2}})}^{2}} = M^{2}$

Step 2.2.1.5.5

Simplify.

$\frac{4 P^{2} B R {(A^{\frac{1}{2}})}^{2} {(G^{\frac{1}{2}})}^{2}}{{(X^{\frac{1}{2}})}^{2} {(y^{\frac{1}{2}})}^{2}} = M^{2}$

Step 2.2.1.5.6

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

Tap for more steps...

Step 2.2.1.5.6.1

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

$\frac{4 P^{2} B R A^{\frac{1}{2} \cdot 2} {(G^{\frac{1}{2}})}^{2}}{{(X^{\frac{1}{2}})}^{2} {(y^{\frac{1}{2}})}^{2}} = M^{2}$

Step 2.2.1.5.6.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 2.2.1.5.6.2.1

Cancel the common factor.

$\frac{4 P^{2} B R A^{\frac{1}{2} \cdot 2} {(G^{\frac{1}{2}})}^{2}}{{(X^{\frac{1}{2}})}^{2} {(y^{\frac{1}{2}})}^{2}} = M^{2}$

Step 2.2.1.5.6.2.2

Rewrite the expression.

$\frac{4 P^{2} B R A^{1} {(G^{\frac{1}{2}})}^{2}}{{(X^{\frac{1}{2}})}^{2} {(y^{\frac{1}{2}})}^{2}} = M^{2}$

Step 2.2.1.5.7

Simplify.

$\frac{4 P^{2} B R A {(G^{\frac{1}{2}})}^{2}}{{(X^{\frac{1}{2}})}^{2} {(y^{\frac{1}{2}})}^{2}} = M^{2}$

Step 2.2.1.5.8

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

Tap for more steps...

Step 2.2.1.5.8.1

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

$\frac{4 P^{2} B R A G^{\frac{1}{2} \cdot 2}}{{(X^{\frac{1}{2}})}^{2} {(y^{\frac{1}{2}})}^{2}} = M^{2}$

Step 2.2.1.5.8.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 2.2.1.5.8.2.1

Cancel the common factor.

$\frac{4 P^{2} B R A G^{\frac{1}{2} \cdot 2}}{{(X^{\frac{1}{2}})}^{2} {(y^{\frac{1}{2}})}^{2}} = M^{2}$

Step 2.2.1.5.8.2.2

Rewrite the expression.

$\frac{4 P^{2} B R A G^{1}}{{(X^{\frac{1}{2}})}^{2} {(y^{\frac{1}{2}})}^{2}} = M^{2}$

Step 2.2.1.5.9

Simplify.

$\frac{4 P^{2} B R A G}{{(X^{\frac{1}{2}})}^{2} {(y^{\frac{1}{2}})}^{2}} = M^{2}$

Step 2.2.1.6

Simplify the denominator.

Tap for more steps...

Step 2.2.1.6.1

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

Tap for more steps...

Step 2.2.1.6.1.1

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

$\frac{4 P^{2} B R A G}{X^{\frac{1}{2} \cdot 2} {(y^{\frac{1}{2}})}^{2}} = M^{2}$

Step 2.2.1.6.1.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 2.2.1.6.1.2.1

Cancel the common factor.

$\frac{4 P^{2} B R A G}{X^{\frac{1}{2} \cdot 2} {(y^{\frac{1}{2}})}^{2}} = M^{2}$

Step 2.2.1.6.1.2.2

Rewrite the expression.

$\frac{4 P^{2} B R A G}{X^{1} {(y^{\frac{1}{2}})}^{2}} = M^{2}$

Step 2.2.1.6.2

Simplify.

$\frac{4 P^{2} B R A G}{X {(y^{\frac{1}{2}})}^{2}} = M^{2}$

Step 2.2.1.6.3

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

Tap for more steps...

Step 2.2.1.6.3.1

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

$\frac{4 P^{2} B R A G}{X y^{\frac{1}{2} \cdot 2}} = M^{2}$

Step 2.2.1.6.3.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 2.2.1.6.3.2.1

Cancel the common factor.

$\frac{4 P^{2} B R A G}{X y^{\frac{1}{2} \cdot 2}} = M^{2}$

Step 2.2.1.6.3.2.2

Rewrite the expression.

$\frac{4 P^{2} B R A G}{X y^{1}} = M^{2}$

Step 2.2.1.6.4

Simplify.

$\frac{4 P^{2} B R A G}{X y} = M^{2}$

Step 3

Solve for $y$ .

Tap for more steps...

Step 3.1

Find the LCD of the terms in the equation.

Tap for more steps...

Step 3.1.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$X y, 1$

Step 3.1.2

The LCM of one and any expression is the expression.

$X y$

Step 3.2

Multiply each term in $\frac{4 P^{2} B R A G}{X y} = M^{2}$ by $X y$ to eliminate the fractions.

Tap for more steps...

Step 3.2.1

Multiply each term in $\frac{4 P^{2} B R A G}{X y} = M^{2}$ by $X y$ .

$\frac{4 P^{2} B R A G}{X y} (X y) = M^{2} (X y)$

Step 3.2.2

Simplify the left side.

Tap for more steps...

Step 3.2.2.1

Cancel the common factor of $X y$ .

Tap for more steps...

Step 3.2.2.1.1

Cancel the common factor.

$\frac{4 P^{2} B R A G}{X y} (X y) = M^{2} (X y)$

Step 3.2.2.1.2

Rewrite the expression.

$4 P^{2} B R A G = M^{2} (X y)$

Step 3.2.3

Simplify the right side.

Tap for more steps...

Step 3.2.3.1

Remove parentheses.

$4 P^{2} B R A G = M^{2} X y$

Step 3.3

Solve the equation.

Tap for more steps...

Step 3.3.1

Rewrite the equation as $M^{2} X y = 4 P^{2} B R A G$ .

$M^{2} X y = 4 P^{2} B R A G$

Step 3.3.2

Divide each term in $M^{2} X y = 4 P^{2} B R A G$ by $M^{2} X$ and simplify.

Tap for more steps...

Step 3.3.2.1

Divide each term in $M^{2} X y = 4 P^{2} B R A G$ by $M^{2} X$ .

$\frac{M^{2} X y}{M^{2} X} = \frac{4 P^{2} B R A G}{M^{2} X}$

Step 3.3.2.2

Simplify the left side.

Tap for more steps...

Step 3.3.2.2.1

Cancel the common factor of $M^{2}$ .

Tap for more steps...

Step 3.3.2.2.1.1

Cancel the common factor.

$\frac{M^{2} X y}{M^{2} X} = \frac{4 P^{2} B R A G}{M^{2} X}$

Step 3.3.2.2.1.2

Rewrite the expression.

$\frac{X y}{X} = \frac{4 P^{2} B R A G}{M^{2} X}$

Step 3.3.2.2.2

Cancel the common factor of $X$ .

Tap for more steps...

Step 3.3.2.2.2.1

Cancel the common factor.

$\frac{X y}{X} = \frac{4 P^{2} B R A G}{M^{2} X}$

Step 3.3.2.2.2.2

Divide $y$ by $1$ .

$y = \frac{4 P^{2} B R A G}{M^{2} X}$

Step 4

Set the denominator in $\frac{4 P^{2} B R A G}{M^{2} X}$ equal to $0$ to find where the expression is undefined.

$M^{2} X = 0$

Step 5

Solve for $P$ .

Tap for more steps...

Step 5.1

Divide each term in $M^{2} X = 0$ by $X$ and simplify.

Tap for more steps...

Step 5.1.1

Divide each term in $M^{2} X = 0$ by $X$ .

$\frac{M^{2} X}{X} = \frac{0}{X}$

Step 5.1.2

Simplify the left side.

Tap for more steps...

Step 5.1.2.1

Cancel the common factor of $X$ .

Tap for more steps...

Step 5.1.2.1.1

Cancel the common factor.

$\frac{M^{2} X}{X} = \frac{0}{X}$

Step 5.1.2.1.2

Divide $M^{2}$ by $1$ .

$M^{2} = \frac{0}{X}$

Step 5.1.3

Simplify the right side.

Tap for more steps...

Step 5.1.3.1

Divide $0$ by $X$ .

$M^{2} = 0$

Step 5.2

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$M = \pm \sqrt{0}$

Step 5.3

Simplify $\pm \sqrt{0}$ .

Tap for more steps...

Step 5.3.1

Rewrite $0$ as $0^{2}$ .

$M = \pm \sqrt{0^{2}}$

Step 5.3.2

Pull terms out from under the radical, assuming positive real numbers.

$M = \pm 0$

Step 5.3.3

Plus or minus $0$ is $0$ .

$M = 0$

Step 6

The domain is all values of $P$ that make the expression defined.

Interval Notation:

$(- \infty, 0) \cup (0, \infty)$

Set-Builder Notation:

${P | P \neq 0}$