+9491 From the intersecting chord theorem, we know that.....
XT * TY = AT * TB
The problem tells us that XT = 4, TY = 6 , and AT = 2(TB) .
4 * 6 = 2(TB) * TB
24 = 2(TB)2
Divide both sides by 2 .
12 = (TB)2
Take the positive (since TB is a length) square root of both sides.
√12 = TB
AT = 2(TB)
We know that TB = √12
AT = 2√12
AB = AT + TB
Plug in the values we know for AT and TB .
AB = 2√12 + √12
Combine like terms.
AB = 3√12
We can simplify √12 since √12 = √(2 * 2 * 3) = √(22) * √3
AB = 3(2√3)
AB = 6√3
+2446 I happen to remember what this decimal is represented in a fraction form, but here is the method one can use to convert \(0.\overline{142857}\) into a fraction.
This first step is very simple; just set it equal to a variable. I'll use the normal x as my variable for this example. Therefore, \(0.\overline{142857}=x\)
The next goal is to get the repeating portion into the whole numbers part. It is probably easier showing by example than by explaining in words:
| \(0.\overline{142857}=x\) | Multiply by 1000000 on both sides. |
| \(142857.\overline{142857}=10000000x\) | As you can see, the repeating section is now in whole numbers. Now, subtract both equations from each other. |
| \(142857=999999x\) | Now, divide by 999999 on both sides. |
| \(x=\frac{142857}{999999}\div\frac{142857}{142857}\) | It is probably hard to realize here, but the GCF of the numerator and denominator is 142857. |
| \(x=\frac{1}{7}\) | |
Look at that! \(0.\overline{142857}=\frac{1}{7}\). That's quite a nice fraction.
+2446 Inequalities can be tricky things, so let's solve them. In this problem, we will solve for x in the equation \(x^2-9>0\):
| \(x^2-9>0\) | Add 9 to both sides of the inequality. | ||
| \(x^2>9\) | Take the square root of both sides. | ||
| \(|x|>\sqrt{9}\) | |||
| \(|x|>3\) | The absolute value splits the solutions into 2 inequalities. Solve them separately. | ||
| Divide by -1 on both sides. Remember that doing so flips the inequality sign. That's easy to forget! | ||
| Npw, let's combine this into a compound inequality, if possible. Unfortunately, in this case, it is not. | ||
Therefore, x must either be greater than 3 or less than -3.
+2446 To evaluate \(3^{-1}*3^5\), you must be comfortable with exponent rules.
| \(3^{-1}*3^5\) | Use the exponent rule that\(a^b*a^c=a^{b+c}\)to simplify the expression. |
| \(3^{-1}*3^5=3^{-1+5}=3^4\) | Now, evaluate 3^4. |
| \(3^4=\underbrace{3*3}*\underbrace{3*3}\) | To ease computation, calculate it as the following as multiplication can be done in any order. |
| \(3^4=9*9=81\) | |
