simplify radicals with variables

Front

What is a radical in mathematics?

Back

A radical is an expression that includes a root, such as a square root, cube root, etc. The most common radical is the square root.

Front

How do you simplify a square root of a variable raised to a power?

Back

To simplify a square root of a variable raised to a power, divide the exponent by 2. The integer part becomes the coefficient outside the radical, and the remainder stays inside.

Front

What is the simplified form of \( \sqrt{x^{33}} \)?

Back

The simplified form is \( x^{16} \sqrt{x} \).

Front

How do you simplify the expression \( 5\sqrt{48x^2} \)?

Back

First, simplify \( \sqrt{48} \) to \( 4\sqrt{3} \) and \( \sqrt{x^2} \) to \( x \). Thus, \( 5\sqrt{48x^2} = 5 \cdot 4 \cdot x \cdot \sqrt{3} = 20x\sqrt{3} \).

Front

What is the process for simplifying \( \sqrt{98b} \)?

Back

First, factor 98 into 49 and 2. Then, \( \sqrt{98b} = \sqrt{49} \cdot \sqrt{2} \cdot \sqrt{b} = 7\sqrt{2b} \).

Front

What is the general rule for simplifying radicals with variables?

Back

The general rule is to separate the variable's exponent into even and odd parts. The even part can be taken out of the radical, while the odd part remains inside.

Front

What does it mean to rationalize the denominator?

Back

Rationalizing the denominator means to eliminate any radicals from the denominator of a fraction.

Front

How do you rationalize the denominator of \( \frac{1}{\sqrt{a}} \)?

Back

Multiply the numerator and denominator by \( \sqrt{a} \) to get \( \frac{\sqrt{a}}{a} \).

Front

What is the simplified form of \( \sqrt{a^4b^2} \)?

Back

The simplified form is \( a^2b \) because \( \sqrt{a^4} = a^2 \) and \( \sqrt{b^2} = b \).

Front

How do you combine like radicals?

Back

To combine like radicals, ensure they have the same radicand (the number or expression inside the radical). Then, add or subtract the coefficients.