What are indices in mathematics?

Indices, or exponents, indicate the power to which a number or variable is raised. For example, in 2³, 2 is the base and 3 is the index, meaning 2 is multiplied by itself 3 times, resulting in 8.

What is the law of indices for multiplying powers with the same base?

How do you simplify √(a × b)?

According to the rule of surds, √(a × b) = √a × √b. For example, √(2 × 8) = √2 × √8 = √2 × 2√2 = 2√2.

What is the rule for dividing indices with the same base?

When dividing powers with the same base, subtract the exponents: a^m / a^n = a^(m-n). For instance, 5⁴ / 5² = 5^(4-2) = 5² = 25.

What is a surd?

A surd is an expression that includes a root (like a square root) that cannot be simplified to remove the root. For example, √3 and √5 are surds.

How do you rationalize the denominator of 5/√2?

To rationalize, multiply the numerator and the denominator by √2: (5/√2) × (√2/√2) = 5√2/2.

Combine like terms: x = (2 + 4)√3 = 6√3.

What is the outcome of (√5)²?

The square of a square root returns the original number: (√5)² = 5.

How can you express x²/√x in terms of x?

Using the laws of indices, x²/√x = x²/x^(1/2) = x^(2 - 1/2) = x^(3/2).

If 4^(2x) = 64, find x.

First express 64 as a power of 4: 64 = 4³. Thus, 2x = 3, leading to x = 3/2.

What is the simplified form of √(50)?

To simplify, factor 50: √(50) = √(25×2) = √(25)×√(2) = 5√2.

Calculate the value of (2√3)².

(2√3)² = 2² × (√3)² = 4 × 3 = 12.

If (1/3)²a = 1/9, find the value of a.

Rewrite 1/9 as (1/3)². Therefore, 2a = 2, leading to a = 1.

What is the conjugate of 3√2 + 4√5?

The conjugate is 3√2 - 4√5.

What is the result of (√2 + √3)(√2 - √3)?

This is a difference of squares: (√2)² - (√3)² = 2 - 3 = -1.