Absolute value: Solve absolute value equations with a number outside the absolute value bars

To solve this problem, let's break it down into the following steps:

• Step 1: Recognize that inside the absolute value, we have $-y^2$. Since $y^2$ is non-negative, $-y^2$ is less than or equal to zero.
• Step 2: Apply the absolute value: Since $-y^2 \leq 0$, we use the property $|-a| = -(-a)$, resulting in $\left|-y^2\right| = y^2$.
• Step 3: Apply the outer negative sign, $-\left|-y^2\right|$, which simplifies to $-y^2$.

Therefore, the solution to the problem is correctly identified as $-y^2$.

To solve this problem, we'll follow these steps:

• Step 1: Compute the expression inside the absolute value.
• Step 2: Apply the absolute value.
• Step 3: Apply the negative sign.

Now, let's work through each step:
Step 1: Compute $4^2$.
Since $4^2 = 16$, we have $\lvert 4^2 \rvert = \lvert 16 \rvert$.
Step 2: Apply the absolute value operation. The absolute value of 16 is 16, so $\lvert 16 \rvert = 16$.
Step 3: Apply the negative sign. We then have $-\lvert 16 \rvert = -16$.

Therefore, the solution to the problem is $-16$.

The problem requires us to evaluate the expression $-\left|-18\right|$.

Step 1: Calculate the absolute value of $-18$.

• The absolute value of a number is the positive version of the number if it's negative. Therefore:
• $\left|-18\right| = 18$

Step 2: Apply the negative sign to the result of the absolute value operation.

• We attach the negative sign that is outside the absolute value:
• $-\left|-18\right| = -18$

Therefore, the result of the expression $-\left|-18\right|$ is $-18$.

To solve the given expression $-\left|15\right|$, we need to find the absolute value of $15$, then apply the negative sign.

The absolute value of a number is the non-negative value of that number without regard to its sign.

Thus, $\left|15\right| = 15$.

Now apply the negative sign: $-\left|15\right| = -15$.

Therefore, the answer is $-15$.

In the given expression, $-\left|7\right|$, the absolute value of $7$ is required.

The absolute value, $\left|7\right|$, is $7$ since absolute value denotes a non-negative distance from zero.

Applying the negative sign changes it to $-7$.

The final result, therefore, is $-7$.

To solve the expression $-\left|23\right|$, first find the absolute value of $23$.

The absolute value of a number is the distance between the number and zero on the number line, so $\left|23\right| = 23$.

Then apply the negative sign: $-\left|23\right| = -23$.

Hence, the correct answer is $-23$.

First, calculate the cube of 5: $5^3 = 125$.

Then, apply the absolute value:
since 125 is positive, $\lvert 125 \rvert = 125$.

Finally, apply the negative sign outside the absolute value: $-\lvert 125 \rvert = -125$.

First, calculate the fourth power of 3: $3^4 = 81$.

Then, apply the absolute value:
since 81 is positive, $\lvert 81 \rvert = 81$.

Finally, apply the negative sign outside the absolute value: $-\lvert 81 \rvert = -81$.

The expression has an absolute value and a negative sign outside of the absolute value. When you take the absolute value of $-x^3$, it results in $|x^3|$, which is $x^3$ assuming$x$ is a real number. The negative sign outside the absolute value inverts it back to $-x^3$. Thus, the correct interpretation of the original expression $-\left|-x^3\right|$ is $-x^3$.

The absolute value function $\left|2z\right|$ simply returns $2z$ when $z$ is positive and $-2z$ when$z$ is negative, ensuring the result is non-negative. However, the minus sign outside the absolute value $-\left|2z\right|$ negates the result of the absolute value. Therefore, $-\left|2z\right|$ results in $-2z$ for all $z$. Hence, the original expression evaluates to $-2z$.

The absolute value of $3y^2$ is $3y^2$ itself because $3y^2$ is always non-negative regardless of the value of $y$ since any real number squared is non-negative. The negative sign outside the absolute value indicates that the expression$-\left|3y^2\right|$ evaluates to $-3y^2$.