Index Laws

Question 1

What does the term “index” refer to in mathematics?

Answer 1

An index (or exponent) refers to the number that shows how many times a base number is multiplied by itself. For example, in a^n, n is the index and a is the base.

Question 2

What is the rule for multiplying two terms with the same base but different indices?

Answer 2

When multiplying terms with the same base, add the indices: a^m * a^n = a^(m+n)

Question 3

What is the rule for dividing two terms with the same base but different indices?

Answer 3

When dividing terms with the same base, subtract the indices: a^m ÷ a^n = a^(m-n)

Question 4

What is the result of raising a power to another power?

Answer 4

When raising a power to another power, multiply the indices: (a^m)^n = a^(m*n)

Question 5

What is the value of any base raised to the power of zero?

Answer 5

Any base raised to the power of zero equals 1, as long as the base is not zero: a^0 = 1

Question 6

How do you deal with negative indices?

Answer 6

A negative index represents the reciprocal of the base raised to the positive index: a^(-n) = 1 / a^n

Question 7

What is the rule for distributing an exponent over a product?

Answer 7

When distributing an exponent over a product of terms, apply the exponent to each term in the product: (ab)^n = a^n * b^n

Question 8

What is the rule for distributing an exponent over a quotient?

Answer 8

When distributing an exponent over a quotient, apply the exponent to both the numerator and the denominator: (a/b)^n = a^n / b^n

Question 9

What does a^(1/n) represent?

Answer 9

a^(1/n) represents the nth root of a: a^(1/n) = n√a

Question 10

What is a^(m/n) equivalent to?

Answer 10

a^(m/n) is equivalent to the nth root of a, raised to the power of m: a^(m/n) = (n√a)^m

Question 11

Simplify x^3 * x^4.

Answer 11

Using the multiplication index law, the result is: x^3 * x^4 = x^(3+4) = x^7

Question 12

Simplify y^5 ÷ y^2.

Answer 12

Using the division index law, the result is: y^5 ÷ y^2 = y^(5-2) = y^3

Question 13

Simplify (3^2)^3.

Answer 13

Using the power of a power rule, the result is: (3^2)^3 = 3^(2*3) = 3^6

Question 14

Express 1 / x^3 using a negative index.

Answer 14

Using the negative index rule, this can be written as: 1 / x^3 = x^(-3)

Question 15

Simplify (2xy)^3.

Answer 15

Applying the exponent to both 2, x, and y: (2xy)^3 = 2^3 * x^3 * y^3 = 8x^3y^3