Skill: Simplify numerical expressions involving exponents and roots

To simplify numerical expressions with exponents and roots:

1. Evaluate powers and roots.
2. Perform multiplication or division.
3. Add or subtract from left to right.
4. Pay attention to parentheses and signs.

Worked example

Simplify:

$$(-3)^2 - 2\sqrt{49}+\sqrt[3]{27}$$

Step 1: Evaluate the exponent and roots.

The parentheses mean the entire number $-3$ is squared:

$$(-3)^2=9$$

The principal square root of $49$ is $7$, and the cube root of $27$ is $3$:

$$\sqrt{49}=7 \qquad\text{and}\qquad \sqrt[3]{27}=3$$

Substitute these values:

$$9-2(7)+3$$

Step 2: Multiply.

$$2(7)=14$$

So the expression becomes:

$$9-14+3$$

Step 3: Add and subtract from left to right.

$$9-14=-5$$ $$-5+3=-2$$

Therefore,

$$\boxed{-2}$$

Remember the difference between $(-3)^2$ and $-3^2$:

$$(-3)^2=9 \qquad\text{but}\qquad -3^2=-(3^2)=-9$$

Parentheses determine whether the negative sign is included in the exponent.