simplify sqrt(-2)*sqrt(-8) — step-by-step solution

Let's reduce:

\sqrt{-2} \times \sqrt{-8}

ⓘ In ℂ, the principal square root of a negative real number is i times the square root of its absolute value, because i² = -1:

\sqrt{-2}

i \times \sqrt{2}

\sqrt{-8}

i \times 2 \times \sqrt{2}

i \times \sqrt{2} \times i \times 2 \times \sqrt{2}

ⓘ Multiplying powers with the same base adds the exponents:

i \times \sqrt{2} \times i \times 2 \times \sqrt{2}

i^{2} \times 2^{\frac{1}{2} + 1 + \frac{1}{2}}

i^{2} \times 2^{\frac{1}{2} + 1 + \frac{1}{2}}

ⓘ i^2 = -1, so the powers of i reduce to 1, i, -1 or -i:

i^{2}

-1

■To add fractions, put them over the same denominator, then add the numerators:

\frac{1}{2} + 1

\frac{1 + 2}{2}

■Multiplying by -1 gives the opposite:

-1 \times 2^{\frac{1 + 2}{2} + \frac{1}{2}}

- 2^{\frac{1}{2} + \frac{1 + 2}{2}}

- 2^{\frac{1}{2} + \frac{1 + 2}{2}}

ⓘ The numerator and the denominator are computed before the division:

1 + 2

3

- 2^{\frac{1}{2} + \frac{3}{2}}

ⓘ To add fractions, put them over the same denominator, then add the numerators:

\frac{1}{2} + \frac{3}{2}

\frac{1 + 3}{2}

- 2^{\frac{1 + 3}{2}}

ⓘ The numerator and the denominator are computed before the division:

1 + 3

4

- 2^{\frac{4}{2}}

ⓘ We simplify the fraction by dividing the numerator and the denominator by

2

, their greatest common divisor:

\frac{4}{2}

2

- 2^{2}

ⓘ Without parentheses, the minus sign is not part of the power■An integer power is a repeated product:

- 2^{2}

- (2 \times 2)

-4

▶Simplified form:

-4