Question

Use the properties of exponents to simplify the expressions. (a) \(\left(5^{2}\right)\left(5^{3}\right)\) (b) \(\left(5^{2}\right)\left(5^{-3}\right)\) (c) \(\frac{5^{3}}{25^{2}}\) (d) \(\left(\frac{1}{4}\right)^{2} 2^{6}\)

Short answer

(a) The simplified expression is \(5^{5}\). (b) The simplified expression is \(5^{-1}\) or \(\frac{1}{5}\). (c) The simplified expression is \(5^{-1}\) or \(\frac{1}{5}\). (d) The simplified expression is \(4\).

Step-by-step solution

Simplify Part (a)

For the expression \( \left(5^{2}\right)\left(5^{3}\right) \), when the bases are the same (in this case they are both 5) and we are multiplying the terms, we add the exponents: \(5^{2+3} = 5^{5}\).

Simplify Part (b)

For the expression \(\left(5^{2}\right)\left(5^{-3}\right)\), again same base which is a 5, but this time one of the exponents is negative. When multiplying, we still add the exponents: \(5^{2+(-3)}= 5^{-1}\)

Simplify Part (c)

For the expression \(\frac{5^{3}}{25^{2}}\), we can rewrite \(25\) as \(5^{2}\). This means the expression can be rewritten as: \(\frac{5^{3}}{(5^{2})^{2}}\). Here, the rule is to multiply the exponents when they are stacked, resulting in \(\frac{5^{3}}{5^{4}}\). When dividing with the same base, we subtract the exponents: \(5^{3-4} = 5^{-1}\)

Simplify Part (d)

For the expression \(\left(\frac{1}{4}\right)^{2} 2^{6}\), we treat the terms separately. First, \(\left(\frac{1}{4}\right)^{2}\) simplifies to \(\frac{1}{16}\). Next, \(2^{6} = 64\). Now, multiply the terms together to get the final answer: \(\frac{1}{16} \times 64 = 4\).

Key concepts

Simplifying Exponential Expressions

Understanding exponential expressions is foundational in math. Simplifying these expressions involves applying a set of rules to make them more manageable and easier to work with. An essential aspect is recognizing that exponentiation is repeated multiplication; for example, \(5^{3}\) means \(5 \times 5 \times 5\). Simplifying then often means looking for patterns or opportunities to combine like terms.

When you see something like \(a^{m} \times a^{n}\), where \(a\) is any number or variable, and \(m\) and \(n\) are exponents, it simplifies to \(a^{m+n}\), which is much cleaner. This fundamental property of exponents helps turn lengthy problems into solvable equations, especially when dealing with the same base. Remember, the base stays the same, and only the exponents change.

Let’s recap one of the solutions from our original exercise using these principles: \(5^{2}\) times \(5^{3}\) simplifies to \(5^{2+3}\), which then further simplifies to \(5^{5}\), following the rule of adding exponents of the same base during multiplication.

Multiplying Exponents

Building on simplification, multiplying exponents is straightforward if you grasp the underlying rule. When multiplying powers with the same base, sum up the exponents. Consider the example \(x^{a} \times x^{b} = x^{a+b}\), where \(x\) could be any real number. The key is that the bases must be the same for this rule to apply.

In our exercise case, the multiplication \(5^{2}\) times \(5^{3}\) falls directly under this rule. This is why the solution simplifies to \(5^{5}\), simply by adding the exponents together. It’s a neat and quick calculation once you remember the rule and apply it automatically whenever you encounter multiplying exponents of the same base.

Negative Exponents

Negative exponents can be intimidating at first, but they follow a simple principle. A negative exponent means the reciprocal of the base raised to the absolute value of the exponent. For instance, \(a^{-n} = \frac{1}{a^{n}}\), where \(n > 0\).

This means when we come across an expression like \(5^{-1}\), we can interpret it as \(\frac{1}{5}\). In our exercise, for \(5^{2}\) times \(5^{-3}\), we apply the multiplication rule adding the exponents, resulting in \(5^{-1}\), which simplifies to \(\frac{1}{5}\). Understanding negative exponents as reciprocals lets you transform them into a more familiar positive exponent form, which often makes solving the rest of the problem much easier.

Dividing Exponents

Dividing exponents comes with its own set of rules, distinct yet related to multiplying exponents. When dividing powers with the same base, subtract the exponent in the denominator from the exponent in the numerator: \(\frac{a^{m}}{a^{n}} = a^{m-n}\). If you end up with a negative exponent, remember the rule for negative exponents.

In the textbook example, when dividing \(\frac{5^{3}}{5^{4}}\), we simply subtract the exponents to get \(5^{-1}\), or \(\frac{1}{5}\) once we apply the negative exponent principle. This operation illustrates how understanding dividing exponents is vital, especially when simplifying expressions with variables in both the numerator and denominator. Applying this rule often leads to more straightforward, easier-to-manage equations.