91Ó°ÊÓ

Absolute values represent the distance a number is from zero on a number line, regardless of direction. This means absolute values are always non-negative. When simplifying expressions involving variables, absolute values ensure that even if the variable could be negative, the result remains non-negative.
For instance, in the exercise, after finding the square root of 49 and simplifying \( x^2 \), we end up with \( 7x \.\). Using the absolute value \( |x| \), we write the result as \( 7|x| \).
Similarly, for \( -\sqrt{81 x^{18}} \), we simplify it to \( -9x^9 \.\) However, we use absolute values to ensure non-negative results, writing \( -9|x^9| \).

Absolute values are crucial in simplifying roots and powers because they guarantee the final answer is correct, even when variables might be negative.

Square roots are fundamental in algebra and indicate a number that, when multiplied by itself, gives the original number. For example, the square root of 49 is 7 because \( 7 \times 7 = 49 \.\) When simplifying expressions with square roots:

• Separate constants and variables.
• Simplify each part individually.

In the exercise
\( \sqrt{49 x^{2}} \), we separate it into \( \sqrt{49} \) and \( \sqrt{x^{2}} \). Simplifying these, we get \( 7 \) and \( |x| \). Hence, \( \sqrt{49 x^{2}} = 7|x| \).

Similarly, with \( \sqrt{81 x^{18}} \), we separate it into constants and variables, i.e., \( \sqrt{81} \) and \( \sqrt{x^{18}} \). We get \( 9 \) and \( x^9 \), so \( -\sqrt{81 x^{18}} = -9|x^9| \).

Breaking down square roots simplifies the process and helps manage complex expressions.

Variable exponents occur regularly in algebra and are simplified using basic exponent rules. When variable exponents are under a square root, divide the exponent by 2. For instance:
In the exercise \( \sqrt{x^{2}} = x \), since the square root and exponent of 2 cancel out each other, we get \( x \).
Now consider, \( \sqrt{x^{18}} = x^9 \) because dividing 18 by 2 gives \( 9 \). This applies similarly to all exponents: \( \sqrt{x^{n}} = x^{n/2} \).
Combining variable exponents with constants:

• Separate constants and variables.
• Simplify each part.

Example in task,
\( \sqrt{49 x^{2}} = 7|x| \).

While solving \( -\sqrt{81 x^{18}} \), we get, \( -9|x^9| \).

Understanding variable exponents and simplification rules are key to handling complex algebraic expressions efficiently.