express powers

Examples of express powers in the following topics:

• Expressed Powers

  • The expressed powers of the President are those expressed in the Constitution of the United States.
  • The expressed powers of the President are those expressed in the Constitution of the United States.
  • Clause 1 states that "the executive Power shall be vested in a President of the United States of America.
  • Perhaps the most important of all presidential powers is commander-in-chief of the United States Armed Forces.
  • Discuss the expressed powers of the President in the Constitution of the United States

• Expressing Functions as Power Functions

  • A power function is a function of the form $f(x) = cx^r$ where $c$ and $r$ are constant real numbers.
  • Polynomials are made of power functions.
  • Since all infinitely differentiable functions can be represented in power series, any infinitely differentiable function can be represented as a sum of many power functions (of integer exponents).
  • Power functions are a special case of power law relationships, which appear throughout mathematics and statistics.
  • Therefore, an arbitrary function that is infinitely differentiable is expressed as an infinite sum of power functions ($x^n$) of integer exponent.

• Power Series

  • A power series (in one variable) is an infinite series of the form:
  • Any polynomial can be easily expressed as a power series around any center $c$, albeit one with most coefficients equal to zero.
  • can be written as a power series around the center $c=1$ as:
  • In such cases, the power series takes the simpler form
  • All power series $f(x)$ in powers of $(x-c)$ will converge at $x=c$.

• Power

  • Legitimate power, power given to individuals willingly by others, is called "authority;" illegitimate power, power taken by force or the threat of force, is called "coercion. " In the corporate environment, power is often expressed as upward or downward.
  • Power can be seen as evil or unjust, but the exercise of power is accepted as endemic to humans as social beings.
  • Because power operates both relationally and reciprocally, sociologists speak of the balance of power between parties to a relationship.
  • All parties to all relationships have some power.
  • Compare the positives and negatives associated with the use of power and how power operates in society

• Simplifying Exponential Expressions

  • Previously, we have applied these rules only to expressions involving integers.
  • The same rule applies to expressions with variables.
  • To simplify the first part of the expression, apply the rule for multiplying two exponential expressions with the same base:
  • We can also apply the rule for raising a power to another exponent:
  • Combining the two terms, our original expression simplifies to $a^5 + 8b^6$.

• Simplifying Radical Expressions

  • Radical expressions containing variables can be simplified to a basic expression in a similar way to those involving only integers.
  • Recall that the $n$th root of a number $x$ is a number $r$ that, when raised to the power of $n$, equals $x$:
  • A radical expression is said to be in simplified form if:
  • There is no factor of the radicand that can be written as a power greater than or equal to the index.
  • Radical expressions that contain variables are treated just as though they are integers when simplifying the expression.

• Logarithms of Powers

  • A simplifying principle of logarithms is that the logarithm of the $p\text{th}$ power of a quantity is $p$ times the logarithm of the quantity.
  • If we apply this rule repeatedly we can device another rule for simplifying expressions of the form $\log_b x^p$.
  • This can be written as $\log_3 (3^x) + \log_3 9 + \log_3(x^{100}) = x+2+100\log_3 x, $ using a combination of the product and power rules.
  • Relate the power rule for logarithms to the rules for operating with exponents, and use this rule to rewrite logarithms of powers

• Concurrent Powers

  • Concurrent powers are the powers that are shared by both the State and the federal government, exercised simultaneously.
  • The United States Constitution affords some powers to the national government without barring them from the states.
  • Concurrent powers are powers that are shared by both the State and the federal government.
  • These concurrent powers including regulating elections, taxing, borrowing money and establishing courts.
  • Describe concurrent powers and how they are exercised in the federal system

• Simplifying Expressions of the Form log_a a^x and a(log_a x)

  • The expressions logaax and alogax can be simplified to x, a shortcut in complex equations.
  • Here, $\log_a{a}$ is asking about what power a must be raised to get a; that power is one.
  • The logarithm of the p-th power of a number is p times the logarithm of the number itself:
  • Because $\log_a{a}=1$, the formula for the logarithm of a power says that for any number x:
  • Once again, $\log_aa^x$ is asking about what power a must be raised to get a; that power is x.

• Introduction to Exponents

  • For example, the expression $b^3$ represents $b \cdot b \cdot b$.
  • Here, the exponent is 3, and the expression can be read in any of the following ways:
  • Now that we understand the basic idea, let's practice simplifying some exponential expressions.
  • Let's look at an exponential expression with 2 as the base and 3 as the exponent:
  • Let's look at another exponential expression, this time with 3 as the base and 5 as the exponent: