Problem 43 - Elementary and Intermediate Algebra

Question

Square or cube each quantity and simplify the result. $$ (\sqrt{7})^{2} $$

Step-by-Step Solution

Answer

The expression \((\sqrt{7})^{2}\) simplifies to 7.

Step 1: Understand what squaring means

Squaring a number or expression is multiplying the number or expression by itself. For example, if you square a number \( x \), you calculate \( x \times x \). For the given expression, \((\sqrt{7})^{2}\), we will multiply \(\sqrt{7}\) by itself.

Step 2: Apply the squaring operation

Calculate \((\sqrt{7})^{2}\). Since squaring a square root cancels the square root, \( (\sqrt{7})^2 = \sqrt{7} \times \sqrt{7} = 7\).

Step 3: Simplification of the expression

As step 2 shows, squaring a square root results in the number under the root itself. Thus, \((\sqrt{7})^{2} = 7\) is already in its simplest form.

Key Concepts

Simplification

In algebra, simplification refers to making an expression easier to understand or work with. It's like cleaning up clutter to see the main idea more clearly.
Simplification involves reducing the complexity of an expression while keeping its original value. Imagine you're organizing a cluttered room, just like decluttering a math problem means breaking it down to its simplest form.

• Group and combine similar terms to reduce excess
• Perform basic arithmetic operations
• Eliminate unnecessary elements

In the exercise \((\sqrt{7})^{2}\), simplifying means converting the expression down to a more straightforward value: 7. Simplifying doesn't change the fundamental essence of the expression, but it makes it more accessible.

Square Roots

Understanding square roots is crucial in solving algebraic expressions involving squaring. A square root of a number is a value that, when multiplied by itself, gives the original number.
For example, the square root of 9 is 3, because 3 x 3 equals 9. The square root symbol \(\sqrt{\quad}\) makes things fun and unique in math. It gives expressions a sense of mystery waiting to be unlocked.

• Square roots simplify by undoing squares
• Key relationship: \(\sqrt{a^{2}} = a\)
• Square roots can exist in positive or negative values, but typically, we stick with the positive, or 'principal' square root

In our example, \(\sqrt{7}\) is the principal square root of 7, an integral part of reaching the final simplified result.

Exponents

Exponents are used to express a number that is multiplied by itself a certain number of times. When you see an exponent, it indicates repeated multiplication. For example, in \(x^2\), the base \(x\) is multiplied by itself once more, making it \(x \times x\).
Let's unravel the mystery further:

• The base is the number being multiplied
• The exponent tells you how many times to multiply the base
• When using exponents with square roots: if the exponent is 2, it 'squares' the root, essentially cancelling it out

In our exercise, \((\sqrt{7})^2\), exponent 2 simplifies \(\sqrt{7}\) to 7, condensing it and making it handier to work with. Learning about exponents unlocks a powerful tool in simplifying algebraic operations, vastly expanding what we can achieve in math. "}]} ]} json_schema создатьAssistant поправитьAssistant добавить что-то Assistant удалить это что-то JSON -- исправлять: исправить добить Assistant добавить создать ответ. JSONschema исправлять JSON -- создать mini_article-expand_chartulareAtcecellentAssistant корректироватьchema включиAssistant изменить# Response Formatsschema {