Question

Exponents and Radicals;

Answer 1

Exponents and Radicals.

This tutorial will be extended and will include the following topics.

Integer Exponents Negative Exponents Roots and Rational Exponents Radicals Rationalizing Denominators and Numerators

Answer 2

Integer Exponents
Non-Negative integers are letters "\(m\)" and "\(n\)". now we develop an analogous property for quotients. By definition,
\[\frac{ 6 }{ 2 }=\frac{ 6\times6\times6\times6\times6 }{ 6\times6}=6\times6\times6=6^3\]
Because there are 5 factors of 6 in the numerator and 2 factors of 6 in the denominator, the quotient has \(5-2=3\) factors of \(6\). In general, we can make the following statement, which applies to any real \(a\) and non-negative integers \(m\) and \(n\) with \(m>n\).
Division with Exponents
To divide \(a^m\) by \(a^n\), subtract the exponets (assuming \(a≠0\)):
\[\frac{ a^m }{ a^n }=a^{m-n}\]
Example 1; We are going to compute each of the following.
(\(a\))
\[\frac{ 5^7 }{ 5^4 }=5^{7-4}=5^3\]
(\(b\))
\[\frac{ (-8)^{10} }{ (-8)^{5}}=(-8)^{10-5}=(-8)^5\]
(\(c\))
\[\frac{ (3c)^9}{ (3c)^3 }=(3c)^{9-3}=(3c)^6\]
When an exponent applies to the product of two numbers, such as \((7\times19)^3\), use the definitions carefully. For instance,
\[(7\times19)^3=(7\times19)(7\times19)(7\times19)=7\times7\times7\times19\times19\times19=7^3\times19^3\]
in other words, \((7\times19)^3\)=\(7^3\times19^3\). This is an example of the following fact, which applies to any real numbers \(a\) and \(b\) and any non-negative integer exponet \(n\).

Answer 3

Product to a Power:
To find \((ab)^n\), apply the exponent to \(every\) factor inside the parentheses:
\[(ab)^n=a^nb^n\]

Answer 4

CAUTION; A common mistake is to write an expression such as \((2x)^5\)=\(2^5x^5\)=\(32x^5\).

Anologous conclusions are valid for quotients (where \(a\) and \(b\) are any real numbers with \(b≠0\) and \(n\) is a non-negative integer exponet).
Quotient to a Power
To find \((a/b)^n\), assuming (\(b≠0\)), apply the exponent to both the numerator and the denominator:
\[(\frac{ a}{ b })^n=\frac{ a^n }{ b^n }\]

Answer 5

Example 2; We'll compute each of the following.
\((a)\)
\[(5y)^3=5^3y^3=125y^3\]
\((b)\)
\[(c^2d^3)^4=(c^2)^4(d^3)^4=c^8d^{12}\]
\((c)\)
\[(\frac{ x}{ 2 })^6=\frac{ x^6 }{ 2^6 }=\frac{ x^6 }{ 64 }\]
\((d)\)
\[(\frac{ a^4 }{ b^3 })^3=\frac{ (a^4)^3 }{ (b^3)^3}=\frac{ a^{12} }{ b^9}\]
\((e)\)
\[(\frac{ (rs)^3 }{ r^4 })^2=\]
We will use several of the preceding properties in succession;
\[(\frac{ (rs)^3 }{r^4})^2=(\frac{ (r^3s^3) }{ r^4})^2\] Product to power in numerator.

Answer 6

~Taking a break~

Answer 7

Helpful ty

Answer 8

Ty ty for this