Expressing the Double Sum Algebraically

Oct 13, 2025 by vinooalluri.com 113 views

26. Expressing Algebraically: Double the Sum of p and q

In algebra, we often need to express mathematical expressions in a more concise and simplified form. One common operation is to double the sum of two variables. In this problem, we are asked to express the double sum of $p$ and $q$ algebraically.

Understanding the Problem

To solve this problem, we need to understand what "double the sum" means. When we say "double the sum," we are referring to the operation of multiplying the sum of two numbers by 2. In other words, if we have two numbers, $p$ and $q$ , their sum is $p+q$ . To double this sum, we need to multiply it by 2.

Expressing the Double Sum Algebraically

Now, let's express the double sum of $p$ and $q$ algebraically. We can start by writing the sum of $p$ and $q$ as $p+q$ . To double this sum, we need to multiply it by 2. Therefore, the double sum of $p$ and $q$ can be expressed as:

2(p+q)

This expression can be read as "2 times the sum of $p$ and $q$ ." It is a concise and simplified way to express the double sum of two variables.

Evaluating the Options

Now, let's evaluate the options given in the problem:

A. $p+q \times 2$

This option is incorrect because it is not a valid algebraic expression. The correct way to express the double sum is to multiply the sum by 2, not to multiply $q$ by 2.

B. $2p+q$

This option is incorrect because it is not the double sum of $p$ and $q$ . It is the sum of $2p$ and $q$ , which is not the same thing.

C. $p+q+2$

This option is incorrect because it is not the double sum of $p$ and $q$ . It is the sum of $p$ , $q$ , and 2, which is not the same thing.

D. $2(p+q)$

This option is correct because it is the double sum of $p$ and $q$ . It is a concise and simplified way to express the double sum of two variables.

Conclusion

In conclusion, the correct way to express the double sum of $p$ and $q$ algebraically is $2(p+q)$ . This expression is a concise and simplified way to express the double sum of two variables.

Solving for x: $\sqrt[x]{27}=3$

In this problem, we are asked to solve for $x$ in the equation $\sqrt[x]{27}=3$ . To solve this equation, we need to isolate $x$ and find its value.

Understanding the Problem

To solve this problem, we need to understand the concept of exponents and roots. The expression $\sqrt[x]{27}$ represents the $x$ th root of 27. In other words, it is the number that, when raised to the power of $x$ , equals 27.

Solving the Equation

Now, let's solve the equation $\sqrt[x]{27}=3$ . To do this, we can start by raising both sides of the equation to the power of $x$ . This will eliminate the root and allow us to solve for $x$ .

\left(\sqrt[x]{27}\right)^x=3^x

27=3^x

Now, we can see that $27$ is equal to $3^3$ . Therefore, we can rewrite the equation as:

3^3=3^x

Since the bases are the same, we can equate the exponents:

x=3

Evaluating the Options

Now, let's evaluate the options given in the problem:

A. $x=3$

This option is correct because it is the solution to the equation $\sqrt[x]{27}=3$ .

B. $x=2$

This option is incorrect because it is not the solution to the equation $\sqrt[x]{27}=3$ .

C. $x=4$

This option is incorrect because it is not the solution to the equation $\sqrt[x]{27}=3$ .

D. $x=5$

This option is incorrect because it is not the solution to the equation $\sqrt[x]{27}=3$ .

Conclusion

In conclusion, the correct solution to the equation $\sqrt[x]{27}=3$ is $x=3$ . This is the only option that satisfies the equation.

Discussion

In this discussion, we have explored two algebraic problems. The first problem involved expressing the double sum of $p$ and $q$ algebraically, while the second problem involved solving for $x$ in the equation $\sqrt[x]{27}=3$ . We have seen that the correct way to express the double sum is $2(p+q)$ , and that the correct solution to the equation $\sqrt[x]{27}=3$ is $x=3$ . These problems demonstrate the importance of understanding algebraic expressions and equations, and how they can be used to solve real-world problems.