How do you get rid of the denominator in a fraction?

Multiply both sides of the equation by the denominator, or multiply the numerator and denominator by a value that clears the denominator. If you just need to rewrite the fraction, multiply the top and bottom by the denominator's conjugate or reciprocal. The goal is to make the denominator 1 so the fraction becomes a whole number or simpler expression.

How do I remove a fraction denominator when solving an equation?

Multiply every term on both sides of the equation by the least common denominator (LCD) of all fractions. This cancels each denominator and leaves you with a simpler equation without fractions. Then solve the resulting equation normally.

Can you get rid of a denominator by multiplying by the reciprocal?

Yes, multiplying a fraction by its denominator's reciprocal clears the denominator because the denominator cancels out. For example, (3/4) × (4/1) = 3. This works because multiplying by the reciprocal is equivalent to multiplying by 1 in a different form.

How do you rationalize a denominator with a square root?

Multiply the numerator and denominator by the conjugate of the denominator if it has two terms, or by the square root itself if it has one term. For instance, 1/√2 becomes √2/2 after multiplying by √2/√2. This removes the radical from the denominator.

What is the fastest way to clear denominators in an equation?

Find the least common denominator of all fractions and multiply both sides of the equation by it in one step. This eliminates every denominator at once instead of clearing them one by one. It is the most efficient method for equations with multiple fractions.

How to Get Rid of Denominator in a Fraction: 3 Easy Methods

To get rid of a denominator in a fraction, multiply both sides of the equation by that denominator, or use cross-multiplication when one fraction equals another. Both moves cancel the bottom number or expression, leaving a simpler equation you can solve normally. This is a standard first step in solving rational equations, and it works the same whether the denominator is a plain number like 5 or a variable like x + 2.

The key idea is that division and multiplication are inverse operations. Whatever sits in the denominator is being divided into the numerator, so multiplying the whole equation by it undoes that division.

What Is a Denominator?

A fraction is written as numerator over denominator, like 3/4. The numerator is the top number: it tells you how many parts you have. The denominator is the bottom number: it tells you how many equal parts the whole is cut into. In 3/4, you have 3 of 4 equal pieces.

A denominator can hold more than plain numbers. It can be a variable (x), a sum (x + 2), or a radical (√x). Any of these blocks solving because the unknown is trapped in a position you cannot easily isolate. You cannot just subtract or add your way past a division symbol. You have to undo the division first, which is what clearing the denominator does.

Khan Academy explains denominators as the divisor in a fraction, and how operations on fractions depend on them.

Method 1: Multiply Both Sides

When a single-variable equation has one fraction, multiply every term on both sides by the denominator. This cancels the bottom and leaves an equation with no fractions at all.

Worked example: solve 6/x = 3.

The denominator is x. Multiply both sides by x:

Left side: (6/x) · x = 6
Right side: 3 · x = 3x

Now you have 6 = 3x. Divide both sides by 3 to get x = 2. Check it: 6/2 = 3. Correct.

The same logic works with an expression denominator. For 5/(x + 1) = 1, multiply both sides by (x + 1):

Left side: 5
Right side: x + 1

So x + 1 = 5, meaning x = 4. Notice you multiplied the entire other side, not just part of it. That is the rule that keeps the equation balanced.

Purplemath stresses that whatever you do to one side of an equation, you must do to the other, which is why the multiplier hits every term.

Method 2: Cross-Multiplication

Cross-multiplication clears denominators when a fraction equals another fraction. You multiply the numerator of the first fraction by the denominator of the second, and the numerator of the second by the denominator of the first.

If a/b = c/d, then a · d = b · c. The denominators vanish in one move.

Worked example: solve 3/x = 6/8.

Multiply across:

3 · 8 = 24
x · 6 = 6x

Set them equal: 24 = 6x. Divide by 6 to get x = 4. Check: 3/4 = 6/8, and both equal 0.75. Correct.

When to Use It

Use cross-multiplication only when each side is a single fraction. If either side has extra terms added or subtracted, switch to the LCD method instead. Trying to cross-multiply a sum like 1/x + 2 = 3/4 gives a wrong result because the left side is not one fraction.

Method 3: LCD for Multiple Fractions

When an equation has several fractions with different denominators, multiply every term by the least common denominator, or LCD. The LCD is the smallest number or expression that every denominator divides into evenly.

Worked example: solve 1/2 + 1/x = 3/4.

The denominators are 2, x, and 4. The LCD is 4x. Multiply every term by 4x:

(4x)(1/2) = 2x
(4x)(1/x) = 4
(4x)(3/4) = 3x

The equation becomes 2x + 4 = 3x. Subtract 2x from both sides: 4 = x. Check: 1/2 + 1/4 = 3/4. Correct.

The LCD method is the most general approach because it handles any number of fractions. Multiply each term, then simplify; the denominators disappear together.

Common Mistakes to Avoid

Clearing denominators is mechanical, but a few errors are common.

Forgetting to distribute. If the denominator or the other side has multiple terms, the multiplier must reach every term. Forgetting one leaves a fraction behind and breaks the balance.
Dropping the denominator’s sign. If the denominator is negative, like -x, multiplying by it flips signs. Keep track of negatives or your answer changes.
Introducing extraneous solutions. Multiplying by an expression containing a variable can create values that make the original denominator zero. Those values are not real solutions and must be rejected after checking. For example, in 1/(x - 2) = 4/(x - 2), x = 2 makes both denominators zero, so it is excluded.
Cross-multiplying non-fraction sums. As noted, this only works when each side is one fraction.

Always substitute your answer back into the original equation. If a denominator becomes zero, discard that solution.

Clearing a Denominator Containing a Radical

A radical in the denominator, such as √x, behaves like any other denominator once you recognize it. Multiply both sides by that radical to remove it. For instance, solve 10/√x = 5. Multiply both sides by √x to get 10 = 5√x. Divide by 5: 2 = √x. Square both sides to get x = 4. Verify that √4 = 2, so 10/2 = 5. Correct.

The extra step of squaring can introduce extraneous solutions, so confirmation is mandatory. If you had obtained a negative value under the radical, you would discard it.

Choosing the Right Method Quickly

Each situation points to one method. If there is a single fraction, multiply both sides by its denominator. If a fraction equals a fraction, cross-multiply. If several different denominators appear in one equation, use the LCD. If a radical sits on the bottom, multiply by the radical. Recognizing the shape of the problem first prevents wasted steps and keeps the algebra short.

Key Takeaways

To clear a denominator, multiply both sides of the equation by it; this undoes the division and cancels the bottom.
Cross-multiplication (a/b = c/d gives a·d = b·c) is the fastest route when one fraction equals another.
For several different denominators, multiply every term by the LCD to remove them all at once.
Multiply the entire opposite side, not just part of it, to keep the equation balanced.
Check every solution in the original equation; any value that makes a denominator zero is extraneous and must be rejected.