Multiplication and Division of Radicals
Multiplication & Division of Radicals
Multiplication of Radicals Recall the following rule of distribution:
n
n
a b =
n
ab
x a × y b = xy ab n
n
n
We can multiply radicals with the same index by multiplying the radicands and coefficients, while keeping the index the same.
Multiplication of Radicals We can multiply radicals with the same index by multiplying the radicands and coefficients, while keeping the index the same.
x 6 =
6x
4 2 × 5 5 = 20 10 3
3
3
Perform the arithmetic in the following expression. Express your answer in simplest radical form. 3
3
3
2 12 × x 12 × 12
Perform the arithmetic in the following expression. Express your answer in simplest radical form.
4 6x (3 2x + 2 6 )
Division of Radicals – One Term Recall: When reducing a fractional radical to its simplest radical form, we removed the radical from the denominator by (1) reducing the radicands, and (2) multiplying the numerator and denominator of the fraction by the radical in the denominator, and simplifying.
Division of Radicals – One Term
This is simply one-term division of radicals.
2 72
= = =
2 2×2×2×3×3 2 2 × 6 2 2
2 2 12
=
2 6
=
2 2×6×6 We follow a similar process when dividing by a two-term expression where at least one term is a radical.
Division of Radicals – Two Terms Step 1: If possible, reduce the radicands.
4 8–2
= =
4 2×2×2 – 2 4 2 2–2
Division of Radicals – Two Terms Step 2: Multiply the numerator and denominator by the expression in the denominator with the opposite sign, and simplify.
4 8–2
This will remove radicals from the denominator:
= = =
4 𝒂+ 𝒃 2×2×2 – 2 4 2 2+2 × 2 2–2 2 2+2
8 2+8 4
=
2 2+2
𝒂− 𝒃 = 𝒂−𝒃
Perform the arithmetic in the following expression. Express your answer in simplest radical form.
3 2 5 + 12
Class Problems: Perform the arithmetic in the following expressions. Express your answers in simplest radical form.
4 2 × 3
( 3 +
6 ×2 3 = 3
8 + 48 3– 3
3
3
16 )( 4 – 2 6 ) = =
Multiplication of Radicals Recall the following rule of distribution:
n
n
a b =
n
ab
x a × y b = xy ab n
n
n
We can multiply radicals with the same index by multiplying the radicands and coefficients, while keeping the index the same.
Multiplication of Radicals We can multiply radicals with the same index by multiplying the radicands and coefficients, while keeping the index the same.
x 6 =
6x
4 2 × 5 5 = 20 10 3
3
3
Perform the arithmetic in the following expression. Express your answer in simplest radical form. 3
3
3
2 12 × x 12 × 12
Perform the arithmetic in the following expression. Express your answer in simplest radical form.
4 6x (3 2x + 2 6 )
Division of Radicals – One Term Recall: When reducing a fractional radical to its simplest radical form, we removed the radical from the denominator by (1) reducing the radicands, and (2) multiplying the numerator and denominator of the fraction by the radical in the denominator, and simplifying.
Division of Radicals – One Term
This is simply one-term division of radicals.
2 72
= = =
2 2×2×2×3×3 2 2 × 6 2 2
2 2 12
=
2 6
=
2 2×6×6 We follow a similar process when dividing by a two-term expression where at least one term is a radical.
Division of Radicals – Two Terms Step 1: If possible, reduce the radicands.
4 8–2
= =
4 2×2×2 – 2 4 2 2–2
Division of Radicals – Two Terms Step 2: Multiply the numerator and denominator by the expression in the denominator with the opposite sign, and simplify.
4 8–2
This will remove radicals from the denominator:
= = =
4 𝒂+ 𝒃 2×2×2 – 2 4 2 2+2 × 2 2–2 2 2+2
8 2+8 4
=
2 2+2
𝒂− 𝒃 = 𝒂−𝒃
Perform the arithmetic in the following expression. Express your answer in simplest radical form.
3 2 5 + 12
Class Problems: Perform the arithmetic in the following expressions. Express your answers in simplest radical form.
4 2 × 3
( 3 +
6 ×2 3 = 3
8 + 48 3– 3
3
3
16 )( 4 – 2 6 ) = =
