Question

Rewrite \(\sqrt{a^{5}}\) using a single index.

Short answer

Question: Rewrite the expression √(a^5) using a single index. Answer: a^(5/2)

Step-by-step solution

Identify the terms inside the square root

We start by recognizing the terms inside the square root, which are \(a^{5}\).

Apply the square root rule

Next, we apply the rule for square roots: \(\sqrt[n]{a^m}=a^{\frac{m}{n}}\). In our case, the expression is \(\sqrt{a^{5}}\) which implies n=2.

Rewrite using a single index

Therefore, we can rewrite \(\sqrt{a^{5}}\) as follows: $$ \sqrt{a^{5}}=a^{\frac{5}{2}} $$ The expression \(\sqrt{a^{5}}\) has been rewritten using a single index as \(a^{\frac{5}{2}}\).

Key concepts

Exponent Rules

Exponent rules are fundamental to working with algebraic expressions. They help us simplify expressions involving powers of the same base. Here are some of the key rules:

• Product of Powers Rule: When multiplying, add the exponents if the bases are the same. For example, \(a^m \times a^n = a^{m+n}\).
• Quotient of Powers Rule: When dividing, subtract the exponents. For example, \(\frac{a^m}{a^n} = a^{m-n}\).
• Power of a Power Rule: When raising a power to another power, multiply the exponents. For example, \((a^m)^n = a^{m \cdot n}\).
• Zero Exponent Rule: Any number to the power of zero is one, \(a^0 = 1\), as long as \(a eq 0\).

Understanding these rules allows us to handle expressions like \(\sqrt{a^5}\), by applying them to rewrite the expression in terms of indices. Numerator and denominator relate to these rules and in applying the root rule.

Square Roots

Square roots are a way of expressing the idea of "where does this number come from when multiplied by itself?" The square root symbol \(\sqrt{}\) asks "what number squared would give us this result?". For example, \(\sqrt{9} = 3\) because \(3^2 = 9\).
The square root is actually a special case of exponentiation. A square root can be expressed using exponents as raising a number to the power of \(\frac{1}{2}\). In mathematical notation, this means \(\sqrt{a} = a^{\frac{1}{2}}\). This rule extends to expressions such as \(\sqrt{a^5}\), leading to expressions like \(a^{\frac{5}{2}}\).
By understanding the connection between roots and exponents, we can tackle equations and expressions involving higher roots or fractional exponents with ease.

Indices

Indices, also known as exponents, represent how many times a number, the base, is multiplied by itself. Indices are written as small numbers to the top-right of the base number.
For example, in \(a^5\), the "5" is the index and represents \(a\) multiplied by itself 5 times: \(a \times a \times a \times a \times a\).
The concept of indices is critical in rewriting expressions like \(\sqrt{a^5}\). Here, the solution involves converting it to a single index by using rules for exponents, resulting in \(a^{\frac{5}{2}}\). This simplifies the operation and provides a clearer view of the underlying mathematics.
Indices are used in scientific notations, calculations in physics, and engineering, showcasing their broad applicability.

Mathematical Notation

Mathematical notation is a standardized way of expressing mathematical ideas in written form. It is essential for communicating complex concepts concisely and accurately.
In the context of the original exercise, mathematical notation allowed us to express a square root of a power, \(\sqrt{a^5}\), in a more simplified single index form, \(a^{\frac{5}{2}}\).

• Superscript: Used for indices, showing repeated multiplication.
• Radical symbol: \(\sqrt{}\) used to denote roots.
• Fractional Exponents: Allows representation of roots as powers, simplifying complex expressions.

Understanding mathematical notation is key in deciphering and solving algebraic expressions efficiently. It helps students navigate through the syntax and semantics of mathematics, equipping them to interpret and solve problems effectively.