Question

Simplify the expression. \(\left(5^{1 / 2} \cdot 5^{1 / 4}\right)\)

Short answer

The simplified form of the given expression is \(5^{3 / 4}\).

Step-by-step solution

Understand the Law of Exponents

The problem can be solved using the law of exponents that states that to multiply like bases, the exponents should be added. It's denoted as \(a^m \cdot a^n = a^{m+n}\). In this case, the base is 5 and the exponents are \(1 / 2\) and \(1 / 4\).

Apply the Law of Exponents

Applying the law of exponents, we get \(5^{1 / 2} \cdot 5^{1 / 4} = 5^{1 / 2 + 1 / 4}\). This step is crucial as it helps to simplify the expression by combining the like bases by addition of their exponents.

Simplification

Further simplifying the exponent, we get \(5^{3 / 4}\). Adding the fractions, \(1 / 2 + 1 / 4 = 2 / 4 + 1 / 4 = 3 / 4\). So, the final simplified form of the given expression is \(5^{3 / 4}\).

Key concepts

Law of Exponents

When working with exponents, understanding the law of exponents is crucial. This rule allows us to perform operations on exponential expressions efficiently. Essentially, when you multiply two expressions with the same base, you simply add their exponents together. The mathematical representation of this law is \( a^m \cdot a^n = a^{m+n} \).

For example, if you have \( 2^3 \cdot 2^4 \) the base, which is 2, remains constant while the exponents are added: \( 3 + 4 = 7 \). Thus, the resultant expression would be \( 2^7 \). Understanding this principle helps us to simplify expressions and solve algebraic problems involving exponents.

Like Bases

Working with like bases means dealing with expressions that have the same base number raised to different powers, such as \( 3^2 \cdot 3^5 \). Since they share the same base (in this case, 3), we are able to apply the law of exponents.

The key to simplifying expressions with like bases is to recognize that only the exponents change, while the base remains untouched. This uniformity is what allows the addition or other operations to be solely on the exponents. Keep in mind that this principle applies only to multiplication and division, where the bases are exactly alike.

Adding Exponents

Adding exponents is a straightforward process when the bases of the expressions being multiplied are the same. As seen in the law of exponents, to add exponents, you align the bases and perform the addition solely on the exponents.

For example, \( 4^{1/3} \cdot 4^{2/3} \) becomes \( 4^{1/3 + 2/3} = 4^1 = 4 \). The simplicity of this rule belies its power to solve complex problems by providing a clear pathway to the simplification of exponential expressions. This operation underscores the importance of understanding fractions since exponents are often represented as fractions in more advanced problems.

Exponential Expressions

Exponential expressions are mathematical expressions where a number (the base) is raised to a certain power (the exponent). This notation represents repeated multiplication of the base. For instance, \( 5^3 \) tells you to multiply 5 by itself 3 times: \( 5 \cdot 5 \cdot 5 \).

Simplifying exponential expressions often involves applying laws of exponents, like adding, subtracting, multiplying or dividing them, and sometimes even using more advanced techniques such as exponential decay or growth formulas. As we have seen in the exercise provided, simplifying such expressions boils down to understanding the underlying principles and clearly applying them step by step.