Question

Write your answer as a power or as a product of powers. $$ \left(2^{3}\right)^{2} $$

Short answer

The simplified answer as a power is \(2^{6}\)

Step-by-step solution

Apply the power to a power rule

The power of a power rule states that, \((a^m)^n = a^{mn}\). Using this rule, simplify \(\left(2^{3}\right)^{2}\) to be \(2^{3 \cdot 2}\).

Simplify the exponent

Calculate the multiplication in the exponent, \(3 \cdot 2 = 6\), leading to the result of \(2^{6}\).

Key concepts

Exponents

Exponents are a shorthand notation for denoting the repeated multiplication of a number by itself. In other words, an exponent indicates how many times you should multiply the base number by itself. For instance, when we see the expression \(2^3\), it means we multiply the base, 2, by itself three times: \(2 \times 2 \times 2\), resulting in 8.

In the context of the power to a power rule, when have an expression like \((a^n)^m\), we are essentially saying, 'Take the number a, multiply it by itself n times to get a power, then take that entire power and multiply it by itself m times.' However, this would be a lengthy process. Thus, mathematics offers a simpler way to compute this with the power to a power rule, turning it into \(a^{n \times m}\).

When teaching exponents, it’s crucial to emphasize the importance of the base and the exponent. The base is the number being multiplied, and the exponent tells us how many times to do the multiplication.

Simplifying Expressions

Simplifying expressions is all about making complex mathematical expressions more manageable and easily understandable. The aim is to write them in the simplest form possible, while retaining the same value. This not only makes calculations easier but also helps to better visualize and grasp mathematical concepts.

In the given exercise, simplifying the expression involves applying the power to a power rule, which reduces the complex expression \((2^3)^2\) to a simpler form, \(2^6\). The process involves less steps and presents the expression in an uncluttered manner. Simplification is a common requirement in algebra and other areas of mathematics, as it allows for clearer communication and potentially reveals more insights into the nature of the problem being solved.

It is helpful for students to learn different rules and properties of algebra that enable simplification, such as the distributive law, combining like terms, and, as demonstrated, the power to a power rule.

Mathematical Properties

Mathematical properties are fundamental truths in mathematics that apply regardless of the numbers or expressions involved. These properties are the underlying tools that make it possible to manipulate and solve algebraic expressions, equations, and other mathematical statements. Some of the key properties include the commutative, associative, and distributive properties.

The power to a power rule is based on these foundational properties, specifically the associative property of multiplication. It states that when you have multiple values to multiply together, you can change the grouping of the numbers and the result will still be the same. The rule allows us to combine the exponents in an elegant and logical way by multiplying them.

Understanding and applying these properties correctly is crucial for students as it builds a strong mathematical foundation. This becomes invaluable as students progress to higher-level math where these properties are applied in more complex situations and prove essential in finding solutions and proving theorems.