Simplify the following Surds: a) Simplify 6/(2sqrt(2)−1). b) Simplify (2sqrt(3)+2)/(2sqrt(3)−2).

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Here are the solutions to the problems:

Q1: Simplify the following Surds.

a) Simplify $\frac{6}{2\sqrt{2}-1}$ .

Rationalize the denominator by multiplying the numerator and denominator by the conjugate of the denominator, which is $2\sqrt{2}+1$ .
$\frac{6}{2\sqrt{2}-1} \times \frac{2\sqrt{2}+1}{2\sqrt{2}+1}$
Multiply the numerators and denominators.
$\frac{6(2\sqrt{2}+1)}{(2\sqrt{2}-1)(2\sqrt{2}+1)}$
Expand the numerator and use the difference of squares formula $(a-b)(a+b)=a^2-b^2$ for the denominator.
$\frac{12\sqrt{2}+6}{(2\sqrt{2})^2 - 1^2}$
Simplify the denominator.
$\frac{12\sqrt{2}+6}{4 \times 2 - 1} = \frac{12\sqrt{2}+6}{8 - 1} = \frac{12\sqrt{2}+6}{7}$
The simplified expression is:
$\boxed{\frac{12\sqrt{2}+6}{7}}$
b) Simplify $\frac{2\sqrt{3}+2}{2\sqrt{3}-2}$ .
Rationalize the denominator by multiplying the numerator and denominator by the conjugate of the denominator, which is $2\sqrt{3}+2$ .
$\frac{2\sqrt{3}+2}{2\sqrt{3}-2} \times \frac{2\sqrt{3}+2}{2\sqrt{3}+2}$
Multiply the numerators and denominators.
$\frac{(2\sqrt{3}+2)(2\sqrt{3}+2)}{(2\sqrt{3}-2)(2\sqrt{3}+2)}$
Expand the numerator using $(a+b)^2=a^2+2ab+b^2$ and the denominator using $(a-b)(a+b)=a^2-b^2$ .
$\frac{(2\sqrt{3})^2 + 2(2\sqrt{3})(2) + 2^2}{(2\sqrt{3})^2 - 2^2}$
Simplify the terms.
$\frac{4 \times 3 + 8\sqrt{3} + 4}{4 \times 3 - 4} = \frac{12 + 8\sqrt{3} + 4}{12 - 4}$
Combine like terms.
$\frac{16 + 8\sqrt{3}}{8}$
Factor out the common term from the numerator and simplify.
$\frac{8(2 + \sqrt{3})}{8} = 2 + \sqrt{3}$
The simplified expression is:
$\boxed{2+\sqrt{3}}$
Q2: Simplify the following Logarithms.

a) Solve $\log_4 (2x+3) - 3 = \log_4 x$ .
Rearrange the equation to group the logarithm terms on one side.
$\log_4 (2x+3) - \log_4 x = 3$
Use the logarithm property $\log_b A - \log_b B = \log_b \left(\frac{A}{B}\right)$ .
$\log_4 \left(\frac{2x+3}{x}\right) = 3$
Convert the logarithmic equation to an exponential equation.
If $\log_b A = C$ , then $A = b^C$ .
$\frac{2x+3}{x} = 4^3$
Calculate $4^3$ .
$\frac{2x+3}{x} = 64$
Multiply both sides by $x$ .
$2x+3 = 64x$
Subtract $2x$ from both sides.
$3 = 64x - 2x$ $3 = 62x$
Solve for $x$ .
$x = \frac{3}{62}$
The solution for $x$ is:
$\boxed{x = \frac{3}{62}}$
b) If $\log_2 y = 3 - \log_2 4^{3/2}$ , find the value of $y$ .
Simplify the term $4^{3/2}$ .
$4^{3/2} = (\sqrt{4})^3 = 2^3 = 8$
Substitute this value back into the equation.
$\log_2 y = 3 - \log_2 8$
Evaluate $\log_2 8$ .
Since $2^3 = 8$ , $\log_2 8 = 3$ .
$\log_2 y = 3 - 3$ $\log_2 y = 0$
Convert the logarithmic equation to an exponential equation.
$y = 2^0$
Calculate $2^0$ .
$y = 1$
The value of $y$ is:
$\boxed{y=1}$

Simplify the following Surds: a) Simplify 6/(2sqrt(2)-1). b) Simplify (2sqrt(3)+2)/(2sqrt(3)-2).

Step-by-step solution

Rationalize the denominator by multiplying the numerator and denominator by the conjugate of the denominator, which is $2\sqrt{2}+1$ .

Multiply the numerators and denominators.

Expand the numerator and use the difference of squares formula $(a-b)(a+b)=a^2-b^2$ for the denominator.

Simplify the denominator.

Rationalize the denominator by multiplying the numerator and denominator by the conjugate of the denominator, which is $2\sqrt{3}+2$ .

Multiply the numerators and denominators.

Expand the numerator using $(a+b)^2=a^2+2ab+b^2$ and the denominator using $(a-b)(a+b)=a^2-b^2$ .

Simplify the terms.

Combine like terms.

Factor out the common term from the numerator and simplify.

Rearrange the equation to group the logarithm terms on one side.

Use the logarithm property $\log_b A - \log_b B = \log_b \left(\frac{A}{B}\right)$ .

Convert the logarithmic equation to an exponential equation.

Calculate $4^3$ .

Multiply both sides by $x$ .

Subtract $2x$ from both sides.

Solve for $x$ .

Simplify the term $4^{3/2}$ .

Substitute this value back into the equation.

Evaluate $\log_2 8$ .

Convert the logarithmic equation to an exponential equation.

Calculate $2^0$ .

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More Mathematics Questions

Simplify the following Surds: a) Simplify 6/(2sqrt(2)-1). b) Simplify (2sqrt(3)+2)/(2sqrt(3)-2).

Step-by-step solution

Rationalize the denominator by multiplying the numerator and denominator by the conjugate of the denominator, which is 22+12\sqrt{2}+122​+1.

Multiply the numerators and denominators.

Expand the numerator and use the difference of squares formula (a−b)(a+b)=a2−b2(a-b)(a+b)=a^2-b^2(a−b)(a+b)=a2−b2 for the denominator.

Simplify the denominator.

Rationalize the denominator by multiplying the numerator and denominator by the conjugate of the denominator, which is 23+22\sqrt{3}+223​+2.

Multiply the numerators and denominators.

Expand the numerator using (a+b)2=a2+2ab+b2(a+b)^2=a^2+2ab+b^2(a+b)2=a2+2ab+b2 and the denominator using (a−b)(a+b)=a2−b2(a-b)(a+b)=a^2-b^2(a−b)(a+b)=a2−b2.

Simplify the terms.

Combine like terms.

Factor out the common term from the numerator and simplify.

Rearrange the equation to group the logarithm terms on one side.

Use the logarithm property log⁡bA−log⁡bB=log⁡b(AB)\log_b A - \log_b B = \log_b \left(\frac{A}{B}\right)logb​A−logb​B=logb​(BA​).

Convert the logarithmic equation to an exponential equation.

Calculate 434^343.

Multiply both sides by xxx.

Subtract $2x$ from both sides.

Solve for xxx.

Simplify the term 43/24^{3/2}43/2.

Substitute this value back into the equation.

Evaluate log⁡28\log_2 8log2​8.

Convert the logarithmic equation to an exponential equation.

Calculate 202^020.

Still stuck on this one?

More Mathematics Questions

Rationalize the denominator by multiplying the numerator and denominator by the conjugate of the denominator, which is $2\sqrt{2}+1$ .

Expand the numerator and use the difference of squares formula $(a-b)(a+b)=a^2-b^2$ for the denominator.

Rationalize the denominator by multiplying the numerator and denominator by the conjugate of the denominator, which is $2\sqrt{3}+2$ .

Expand the numerator using $(a+b)^2=a^2+2ab+b^2$ and the denominator using $(a-b)(a+b)=a^2-b^2$ .

Use the logarithm property $\log_b A - \log_b B = \log_b \left(\frac{A}{B}\right)$ .

Calculate $4^3$ .

Multiply both sides by $x$ .

Solve for $x$ .

Simplify the term $4^{3/2}$ .

Evaluate $\log_2 8$ .

Calculate $2^0$ .