Evaluate ( square root of 3/2)^2

Solution:

Step:1

Express $\sqrt{\frac{3}{2}}$ as $\frac{\sqrt{3}}{\sqrt{2}}$ and then square it: ${\left(\frac{\sqrt{3}}{\sqrt{2}}\right)}^2$.

Step:2

Rationalize the denominator by multiplying $\frac{\sqrt{3}}{\sqrt{2}}$ with $\frac{\sqrt{2}}{\sqrt{2}}$: ${(\frac{\sqrt{3}}{\sqrt{2}}\cdot \frac{\sqrt{2}}{\sqrt{2}})}^2$.

Step:3

Simplify the denominator step by step.

Step:3.1

Multiply the numerators and denominators: ${\left(\frac{\sqrt{3}\cdot \sqrt{2}}{\sqrt{2}\cdot \sqrt{2}}\right)}^2$.

Step:3.2

Express $\sqrt{2}$ as a power of 2: ${\left(\frac{\sqrt{3}\cdot \sqrt{2}}{(2^{\frac{1}{2}}{)}^1\cdot \sqrt{2}}\right)}^2$.

Step:3.3

Apply the power rule for exponents: ${\left(\frac{\sqrt{3}\cdot \sqrt{2}}{(2^{\frac{1}{2}}{)}^1\cdot (2^{\frac{1}{2}}{)}^1}\right)}^2$.

Step:3.4

Combine the exponents using the rule $a^m\cdot a^n=a^{m+n}$: ${\left(\frac{\sqrt{3}\cdot \sqrt{2}}{(2^{\frac{1}{2}}{)}^{1+1}}\right)}^2$.

Step:3.5

Add the exponents: ${\left(\frac{\sqrt{3}\cdot \sqrt{2}}{(2^{\frac{1}{2}}{)}^2}\right)}^2$.

Step:3.6

Rewrite $(2^{\frac{1}{2}}{)}^2$ as 2.

Step:3.6.1

Represent $\sqrt{2}$ as $2^{\frac{1}{2}}$: ${\left(\frac{\sqrt{3}\cdot \sqrt{2}}{(2^{\frac{1}{2}}{)}^2}\right)}^2$.

Step:3.6.2

Apply the power rule $(a^m{)}^n=a^{mn}$: ${\left(\frac{\sqrt{3}\cdot \sqrt{2}}{2^{\frac{1}{2}\cdot 2}}\right)}^2$.

Step:3.6.3

Multiply the exponents: ${\left(\frac{\sqrt{3}\cdot \sqrt{2}}{2^{\frac{2}{2}}}\right)}^2$.

Step:3.6.4

Simplify the exponent: ${\left(\frac{\sqrt{3}\cdot \sqrt{2}}{2^1}\right)}^2$.

Step:3.6.5

Square the fraction: ${\left(\frac{\sqrt{3}\cdot \sqrt{2}}{2}\right)}^2$.

Step:4

Combine the radicals in the numerator: ${\left(\frac{\sqrt{3\cdot 2}}{2}\right)}^2$.

Step:5

Simplify the expression further.

Step:5.1

Square the numerator and denominator separately: $\frac{(\sqrt{6}{)}^2}{2^2}$.

Step:5.2

Rewrite $(\sqrt{6}{)}^2$ as 6.

Step:5.2.1

Express $\sqrt{6}$ as $6^{\frac{1}{2}}$: $\frac{(6^{\frac{1}{2}}{)}^2}{2^2}$.

Step:5.2.2

Apply the power rule $(a^m{)}^n=a^{mn}$: $\frac{6^{\frac{1}{2}\cdot 2}}{2^2}$.

Step:5.2.3

Simplify the exponent: $\frac{6^{\frac{2}{2}}}{2^2}$.

Step:5.2.4

Reduce the fraction: $\frac{6^1}{2^2}$.

Step:5.2.5

Evaluate the exponent: $\frac{6}{2^2}$.

Step:5.3

Square the denominator: $\frac{6}{4}$.

Step:5.4

Reduce the fraction by canceling common factors.

Step:5.4.1

Factor out a 2 from 6: $\frac{2\cdot 3}{4}$.

Step:5.4.2

Cancel the common factors.

Step:5.4.2.1

Factor out a 2 from 4: $\frac{2\cdot 3}{2\cdot 2}$.

Step:5.4.2.2

Cancel the 2s: $\frac{\cancel{2}\cdot 3}{\cancel{2}\cdot 2}$.

Step:5.4.2.3

Rewrite the simplified fraction: $\frac{3}{2}$.

Step:6

Present the final result in various forms.

Exact Form: $\frac{3}{2}$ Decimal Form: $1.5$ Mixed Number Form: $1\frac{1}{2}$

Knowledge Notes:

The problem involves evaluating the square of a square root, which is a common operation in algebra. The steps taken to solve this problem include:

1. Rationalizing the Denominator: This is the process of eliminating radicals from the denominator of a fraction. In this case, we multiply by a fraction equivalent to 1 that contains the radical in both the numerator and denominator.

2. Simplifying Radicals: When radicals (square roots) are multiplied or divided, they can often be combined or simplified using the properties of exponents.

3. Properties of Exponents: The power rule $(a^m{)}^n=a^{mn}$ and the product rule $a^m\cdot a^n=a^{m+n}$ are used to simplify expressions involving exponents.

4. Squaring a Fraction: When squaring a fraction, both the numerator and the denominator are squared separately.

5. Reducing Fractions: Common factors in the numerator and denominator can be canceled to simplify the fraction.

6. Alternative Forms: The final result can be expressed in different forms, such as an exact fraction, a decimal, or a mixed number, depending on the context or preference.