91Ó°ÊÓ

When simplifying algebraic expressions, one crucial concept is understanding 'like terms'. Like terms are terms within an expression that have the same variable part. For instance, in the expression given:

$$12 \, \sqrt{a} + 8 \, \sqrt{5b} - 7 \, \sqrt{a} - 17 \, \sqrt{5b}$$

we have the terms \(12 \, \sqrt{a}\) and \(-7 \, \sqrt{a}\) which both contain \(\sqrt{a}\). Therefore, they are like terms. The same applies to \(8 \, \sqrt{5b}\) and \(-17 \, \sqrt{5b}\), which both contain \(\sqrt{5b}\).

By identifying like terms, we can combine them to simplify the expression. This makes working with algebraic expressions more manageable and less overwhelming.

In algebra, a 'coefficient' is the numerical factor of a term that contains a variable. For example, in the term \(12 \, \sqrt{a}\), the number 12 is the coefficient. Coefficients tell us how many times to multiply the variable part.

In the given exercise, each term has a coefficient that needs to be combined with the coefficients of its like terms. So for \(12 \, \sqrt{a}\) and \(-7 \, \sqrt{a}\), we combine the coefficients: \(12 - 7 = 5\). This leaves us with \(5 \, \sqrt{a}\).

Similarly, combining the coefficients for \(8 \, \sqrt{5b}\) and \(-17 \, \sqrt{5b}\), we get: \(8 - 17 = -9\). This results in \(-9 \, \sqrt{5b}\). Understanding coefficients is vital as it allows us to manipulate and simplify expressions accurately and efficiently.

The 'square root' of a number is a value that, when multiplied by itself, gives the original number. For instance, the square root of 16 is 4 because \(4 \times 4 = 16\). In algebra, the square root is symbolized by the radical sign (\(\sqrt{}\)).

In the expression given, we deal with square roots of variables, such as \(\sqrt{a}\) and \(\sqrt{5b}\). These square roots are often combined with coefficients. Understanding square roots is essential when simplifying expressions, especially when identifying and combining like terms.

Simplifying expressions with square roots involves ensuring that the terms under the square root sign (radicands) are identical when combining them. This principle is applied in our exercise to combine terms effectively and simplify the overall expression.