Express 1.5 × 10⁶/2.5 × 10⁻⁴ in the standard form

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Express 1.5 × 10⁶/2.5 × 10⁻⁴ in the standard form.

1.5 × 10⁶/2.5 × 10⁻⁴ standard form is 6 × 10^9.

To express (1.5 × 10^6) / (2.5 × 10^-4) in standard form, we need to simplify the fraction and then represent the result in scientific notation.

First, let’s simplify the fraction:

(1.5 / 2.5) × (10^6 / 10^-4)

Now, divide the numbers and simplify the powers of 10:

(1.5 / 2.5) × 10^(6 – (-4))

(1.5 / 2.5) × 10^(6 + 4)

(0.6) × 10^10

6 × 10^9

So, (1.5 × 10^6) / (2.5 × 10^-4) in standard form is 6 × 10^9.

Inter Conversion between Standard and Normal Forms

In mathematics, particularly in the realm of linear algebra and optimization, there are often discussions about transforming mathematical expressions or equations between standard form and normal form. These concepts are especially prevalent in topics like linear programming and quadratic optimization.

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Standard Form:

  • In linear algebra, standard form means arranging linear equations in the form Ax + By = C, where A, B, and C are constants, and x and y are variables.
  • In quadratic optimization, standard form involves expressing quadratic functions as f(x) = x^T Q x + c^T x + d, where Q is a symmetric matrix, c is a vector, and d is a constant.
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Normal Form:

  • In linear algebra, normal form can refer to row-echelon form or reduced row-echelon form, used in Gaussian elimination.
  • In quadratic optimization, normal form involves expressing quadratic functions as f(x) = (1/2) x^T Px + q^T x + r, where P is a positive definite matrix, q is a vector, and r is a constant.

Interconverting between these forms requires applying mathematical operations to transform the expressions while maintaining their properties and solutions. For instance, converting a linear equation from standard to slope-intercept form involves isolating y. Converting a quadratic function in optimization problems may involve completing the square or other techniques to rewrite the expression.

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Source: Math Hello Kitty
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