Question

For Exercises \(1-74,\) simplify. $$28 \div(-7)+\sqrt{49}+(-3)^{3}$$

Short answer

Question: Simplify the following expression: \(\frac{28}{-7} + \sqrt{49} + (-3)^3\) Answer: -24

Step-by-step solution

Division

Divide \(28\) by \(-7\): \(\frac{28}{-7} = -4\).

Square root

Calculate the square root of \(49\): \(\sqrt{49} = 7\).

Exponentiation

Calculate the cube of \(-3\): \((-3)^3 = -27\).

Combine the results

Add the results from Steps 1, 2, and 3: \(-4 + 7 + (-27) = -4 + 7 - 27 = 3 - 27 = -24\). So, the simplified expression is \(-24\).

Key concepts

Division

In mathematics, division is one of the fundamental operations. It involves splitting a number into equal parts. For example, if you divide 28 by -7, you are essentially asking: how many times does -7 fit into 28?
In our original problem, we have:- **Dividend**: 28- **Divisor**: -7To solve it, you perform the division operation: \[\frac{28}{-7} = -4\]Here, you notice that dividing by a negative number results in a negative quotient.
Remember:

• Dividing a positive by a negative gives a negative result.
• Dividing two negatives gives a positive result.

Square Roots

A square root of a number is a value that, when multiplied by itself, gives the original number. It is represented with the radical symbol \(\sqrt{}\). Understanding square roots helps in simplifying expressions that contain them.
In our exercise, the expression contains \(\sqrt{49}\). This can be broken down as:\[\sqrt{49} = 7\]Here, 7 is the number that gives 49 when multiplied by itself (7 x 7 = 49).
Key points:

• Square roots are only defined for non-negative numbers in real numbers.
• Each positive number has two square roots: one positive and one negative.

Exponentiation

Exponentiation is the process of raising a number to a power. It is depicted by the raise to the power notation. For example, \((-3)^3\) means -3 is used as a factor three times.
Mathematically, \[(-3)^3 = -3 \times -3 \times -3 = -27\]The negative sign remains because you have an odd exponent, and multiplying two negative numbers results in a positive, while an additional multiplication by a negative turns it negative again.
Tips:

• Positive base with an odd exponent results in a positive number.
• Negative base with an even exponent results in a positive number.
• Negative base with an odd exponent results in a negative number.

Negative Numbers

Negative numbers are less than zero and are used to represent values below a defined zero point. When working with negative numbers, certain rules apply that help in calculations.
Useful rules when combining numbers:- Adding a positive and a negative number involves subtracting the smaller absolute value from the larger, and using the sign of the larger.- Subtraction of negative numbers is akin to adding a positive.

• If you subtract a negative number, it’s the same as addition.
• Adding two negative numbers results in a sum that's more negative.

When you combined the results from our steps, you did it in stages:- \(-4 + 7 = 3\)- \(3 + (-27) = 3 - 27 = -24\)This approach ensures clarity in handling negative values.