Method

Simplifying Radical Expressions

Expressions involving radicals be simplified using various properties of exponents and radicals. Consider the following square root.

50 x^{6} y^{4}

Follow these four steps to simplify the expression.

Factor the Radicand

Factor the number in the radicand into its prime factors and the variables into their smallest exponents. In this example, the number under the radical is

50

and it is a product of powers of

2

and

5 .

50 = 2 \cdot 5^{2}

Since

x^{6}

and

y^{4}

are already prime factors, the radical expression can be rewritten as follows.

50 x^{6} y^{4} = 2 \cdot 5^{2} \cdot x^{6} \cdot y^{4}

Separate Perfect Powers

Identify and separate perfect powers that match the index of the radical. Identifying perfect squares for a square root means recognizing expressions that can be rewritten as the product of two identical powers with integer exponents. In the given example,

5^{2},

x^{6},

and

y^{4}

are all perfect squares.

2 \cdot 5^{2} \cdot x^{6} \cdot y^{4} = 2 \cdot 5^{2} \cdot (x^{3})^{2} \cdot (y^{2})^{2}

Simplify the Perfect Powers

The product property of radicals states that for any non-negative numbers the root of their product equals the product of their roots. This allows radicals to be split into separate radicals. Next, simplify each perfect square radical by taking their square roots individually.

2 \cdot 5^{2} \cdot (x^{3})^{2} \cdot (y^{2})^{2}

SqrtProd

$a \cdot b = a \cdot b$

2 \cdot 5^{2} \cdot (x^{3})^{2} \cdot (y^{2})^{2}

SqrtPowToNumber

$a^{2} = a$

2 \cdot 5 \cdot (x^{3})^{2} \cdot (y^{2})^{2}

SqrtToAbs

$a^{2} = ∣ a ∣$

2 \cdot 5 \cdot ∣ x^{3} ∣ \cdot ∣ y^{2} ∣

Because

y^{2}

is always non-negative, the absolute value of

y^{2}

equals

y^{2} .

However, this rule does not hold for

x^{3}

because its exponent is odd.

2 \cdot 5 \cdot ∣ x^{3} ∣ \cdot ∣ y^{2} ∣ = 2 \cdot 5 \cdot ∣ x^{3} ∣ \cdot y^{2}

Combine the Results

Finally, combine the simplified terms outside the root while keeping any terms that are not perfect powers inside the radical.

2 \cdot 5 \cdot ∣ x^{3} ∣ \cdot y^{2} = 52 ∣ x^{3} ∣ y^{2}

Note that, depending on how complex the given radical expression is, different techniques, such as rationalizing or combining radicals, can be applied.

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