/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 51 Expand the expression. $$ (3... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 51

Expand the expression. $$ (3+\sqrt{x})^{2} $$

Short Answer

Expert verified
The short answer based on the given step-by-step solution is: To expand the expression \((3+\sqrt{x})^{2}\), apply the binomial expansion formula \((a+b)^{2} = a^{2} + 2ab + b^{2}\), with \(a=3\) and \(b=\sqrt{x}\). After calculating the individual terms and combining them, the expanded expression is \(9 + 6\sqrt{x} + x\).

Step by step solution

01

Recognize the Binomial

To expand the given expression, we need to recognize it as a binomial raised to a power. The binomial in this case is \((3+\sqrt{x})\) and it is raised to the power of 2.
02

Apply the Binomial Expansion Formula

In order to expand the expression, we will use the formula \((a+b)^{2} = a^{2} + 2ab + b^{2}\), where \(a=3\) and \(b=\sqrt{x}\).
03

Calculate the Individual Terms

Using the formula from step 2, we will now calculate each term: - \(a^{2} = (3)^{2} = 9\) - \(2ab = 2(3)(\sqrt{x}) = 6\sqrt{x}\) - \(b^{2} = (\sqrt{x})^{2} = x\)
04

Combine the Terms

Now, we can combine the terms calculated in step 3 to form the expanded expression: \((3+\sqrt{x})^{2} = 9 + 6\sqrt{x} + x\) The expanded expression is \(9 + 6\sqrt{x} + x\).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Polynomials
Polynomials are mathematical expressions involving a sum of powers of variables with coefficients. They often look like this: \(a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0\). Each part of the polynomial is a term, and it can include variables raised to various powers.
When dealing with polynomials, it's important to recognize terms with variables and their respective powers. For example, in the expression \((3 + \sqrt{x})^2\), after expansion, you have terms like \(9, 6\sqrt{x},\) and \(x\).
  • The constant term here is \(9\).
  • The linear term involving square roots is \(6\sqrt{x}\).
  • The term \(x\) represents a second-degree term.
Understanding these terms is vital in recognizing how complex expressions can be simplified and expanded into polynomials.
Algebra
Algebra is a branch of mathematics that deals with symbols and the rules for manipulating these symbols. It's a way to solve problems with unknowns and express relationships between variables. In our binomial example, algebra helps us expand and simplify the expression \((3+\sqrt{x})^{2}\).
The key steps involve recognizing patterns and applying algebraic formulas. For binomials, a common formula is \((a+b)^2 = a^2 + 2ab + b^2\). Here, \(a = 3\) and \(b = \sqrt{x}\), which are straightforward to substitute into the formula.
  • \(a^2\) means multiplying the first term by itself, resulting in \(9\).
  • For \(2ab\), multiply twice the product of the terms, giving \(6\sqrt{x}\).
  • The term \(b^2\), squaring the second term \(\sqrt{x}\), results in \(x\).
Algebra allows us to see how each component in our expression interacts, helping to create a clear expanded form.
Exponents
Exponents are a shorthand way to show how many times a number is multiplied by itself. In algebra, they're crucial for simplifying expressions, especially in polynomials and binomial expansions.
When you see an expression like \((3+\sqrt{x})^{2}\), the exponent tells us to multiply the binomial by itself twice. Understanding this helps in applying the binomial expansion formula effectively.
  • The exponent \(2\) in the expression denotes squaring, meaning two of \((3 + \sqrt{x})\) are multiplied together.
  • This multiplication leads to using \(a^2, 2ab, \text{and } b^2\), resulting in a clear expansion \(9 + 6\sqrt{x} + x\).
  • Recognizing exponents aids in tackling more complex algebraic equations and understanding relationships involving repeated multiplication.
Using exponents efficiently simplifies complex equations, making algebraic manipulation more manageable and understandable.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Suppose \(m\) is a positive integer. Explain why \(10^{m}\), when written out in the usual decimal notation, is the digit 1 followed by \(m 0^{\prime}\) s.

Evaluate the indicated quantities. Do not use a calculator-pushing buttons for these exercises will not help you understand rational powers. $$ (-8)^{7 / 3} $$

Find all real numbers \(x\) that satisfy the indicated equation. $$ x^{2 / 3}-6 x^{1 / 3}=-8 $$

Find a formula for the inverse function \(f^{-1}\) of the indicated function \(f\). $$ f(x)=x^{9} $$

Suppose \(t\) is an irrational number. Explain why at least one of \(t^{5}\) and \(t^{8}\) is an irrational number.
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Calculus

Read Explanation

Mechanics Maths

Read Explanation

Logic and Functions

Read Explanation

Geometry

Read Explanation

Pure Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.