How do you simplify sqrt{12x^7}?

In summary, the simplification of sqrt{12x^7} is 2|x^3|sqrt{3x}. This is because sqrt{12x^7} can be broken down into 2|x^3|sqrt{3x}, where |x| is the absolute value of x. This is a more proper way of approaching the problem as it takes into account all possible values of x, not just positive values.
  • #1
mathdad
1,283
1
Simplify sqrt{12x^7}

Solution:

sqrt{4•3•x^6•x}

2x^3(sqrt{3x})

Is this correct?

Note 1:

According to wolfram, the answer is 2sqrt{3}•sqrt{x^7}

Note 2:

The term x^3 in my answer comes from the breakdown of sqrt{x^7} as sqrt{x^6•x}. Isn't sqrt{x^6} the same as x^3?
 
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  • #2
Your working assumes $0\le x$...it would be more proper to say:

\(\displaystyle \sqrt{12x^7}=2\left|x^3\right|\sqrt{3x}\)

Because:

\(\displaystyle |x|\equiv\sqrt{x^2}\)
 
  • #3
MarkFL said:
Your working assumes $0\le x$...it would be more proper to say:

\(\displaystyle \sqrt{12x^7}=2\left|x^3\right|\sqrt{3x}\)

Because:

\(\displaystyle |x|\equiv\sqrt{x^2}\)

Back in high school over 30 years ago, my teachers never taught to include the absolute value to a square root problem. For me sqrt{x^2} = x not the |x| but of course, you are right.
 

FAQ: How do you simplify sqrt{12x^7}?

1. What is an algebraic radical?

An algebraic radical is a mathematical expression containing a radical sign (√) and a variable or variables. It is also known as a radical expression.

2. How can we simplify algebraic radicals?

To simplify algebraic radicals, we need to follow certain rules such as finding perfect square factors, multiplying and dividing by rationalizing factors, and combining like terms. The goal is to eliminate the radical (√) and write the expression in its simplest form.

3. What is the difference between rational and irrational radicals?

Rational radicals can be written as a fraction in simplest form, while irrational radicals cannot be expressed as a simplified fraction. Irrational radicals have decimal expansions that do not terminate or repeat, such as √2 = 1.414213... Rational and irrational radicals can be identified by using the rationalizing factor √a x √a = a, where a is a rational number.

4. Can we add or subtract algebraic radicals?

Yes, we can add or subtract algebraic radicals if they have the same radical term and coefficient. For example, √2x + √2x = 2√2x, but √2x + √3x cannot be simplified further. However, we cannot add or subtract irrational radicals as they cannot be written as a simplified fraction.

5. How do we solve equations with algebraic radicals?

To solve equations with algebraic radicals, we need to isolate the radical term on one side of the equation and then square both sides to eliminate the radical. However, we must check for extraneous solutions, which are solutions that do not satisfy the original equation, as squaring both sides can introduce them.

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