Question

1. consider the given diagram where \\( \overleftrightarrow { d b } \\) and \\( \overleftrightarrow { c f } \\) meet at point \\( a \\). point \\( a \\) is also the endpoint of \\( \overrightarrow { a e } \\).

a. describe an angle relationship that would help you solve for \\( x \\).
b. write an equation to find the value of \\( x \\). then determine the measure of \\( \angle e a d \\).

1. which expression is equivalent to \\( 5 ^ { 6 } \cdot 5 ^ { 4 } \\)?

a. \\( 25 ^ { 24 } \\)
b. \\( 10 ^ { 10 } \\)
c. \\( 5 ^ { 24 } \\)
d. \\( 5 ^ { 10 } \\)

1. consider the equation \\( 8 ^ { 15 } \cdot 8 ^ { x } = 8 ^ { 30 } \\). which equation can be used to determine the value of \\( x \\)?

a. \\( 15 + x = 30 \\)
b. \\( 15 - x = 30 \\)
c. \\( 15 \cdot x = 30 \\)
d. \\( 15 \div x = 30 \\)

Explanation:

Question 6

Part a

The sum of angles around a point is \(360^{\circ}\). In the given diagram, the angles \((x + 5)^{\circ}\), \(40^{\circ}\), \(120^{\circ}\) and the straight - line angle (which is \(180^{\circ}\) for the line \(DB\)) can be related. Using the angle - sum property of angles around a point (or the fact that the sum of angles on a straight line and the other given angles can be used). Another way is to note that the sum of angles \((x + 5)^{\circ}+40^{\circ}+120^{\circ}=180^{\circ}\) (since \(DB\) is a straight line and the non - overlapping angles on one side of \(DB\) sum up to \(180^{\circ}\)).

Step1: Write the equation

Since the sum of angles \((x + 5)^{\circ}+40^{\circ}+120^{\circ}=180^{\circ}\) (angles on a straight line \(DB\)).

$$x + 5+40+120=180$$

Step2: Simplify the left - hand side

$$x+(5 + 40+120)=180$$

$$x + 165=180$$

Step3: Solve for \(x\)

Subtract \(165\) from both sides of the equation.

$$x=180 - 165$$

$$x = 15$$

Step4: Find the measure of \(\angle EAD\)

\(\angle EAD=(x + 5)^{\circ}\), substitute \(x = 15\)
\(\angle EAD=(15 + 5)^{\circ}=20^{\circ}\)

Step1: Use the rule of exponents \(a^{m}\cdot a^{n}=a^{m + n}\)

For \(5^{6}\cdot5^{4}\), where \(a = 5\), \(m = 6\) and \(n = 4\)

$$5^{6}\cdot5^{4}=5^{6 + 4}$$

Step2: Calculate the exponent

$$5^{6+4}=5^{10}$$

Answer:

The angles \((x + 5)^{\circ}\), \(40^{\circ}\) and \(120^{\circ}\) are supplementary (they add up to \(180^{\circ}\) as they lie on a straight line \(DB\)).

Part b